So barbarian gives you Strength and Constitution; bear warrior gives you both but more. Assuming you still want to avoid multiclassing, feats are your only real options. Those probably should look something like this:
City Brawler bonus feats: Improved Unarmed Strike, Two-Weapon Fighting
Human bonus feat: Improved Grapple
- Extra Rage
- Power Attack
- Improved Bull Rush
- Shock Trooper
- Improved Two-Weapon Fighting1
- Greater Two-Weapon Fighting1
- Versatile Unarmed Strike2
1 These feats require a lot of Dexterity. It’s entirely possible that you won’t have and won’t want that much Dexterity. In that case... more Extra Rage, I guess.
2 Or anytime earlier if you feel you’re running into a lot of such DR.
That gets you five rages per day, your unarmed strikes can deal a variety of damage types, and at the end you get a lot of attacks in a grapple. But your grappling will be only mediocre, your attacks are not going to do particularly high damage, and Shock Trooper is only thrown in there because it’s a decent feat that will make you at least not entirely useless when you can’t grapple.
Some multiclassing options
If you can multiclass, you should. Barbarian does not offer much at all from 3rd to 7th level. If nothing else, you can take fighter levels to free up more feats to take Extra Rage, so the one significant bonus at Barbarian 4 can actually be done better without barbarian levels.
Fighter
You don’t really need feats all that badly, but several of the feats you’re taking are fighter bonus feats, which means you can take them with a couple levels of fighter and take Extra Rage instead. You can also snag Combat Reflexes if your Dexterity isn’t awful, so you can take an attack of opportunity, and with Improved Grab, start a grapple then-and-there.
But fighter should really never be taken for more than two levels. A feat per level is OK, a feat every other level is awful.
Ranger
The advantage of ranger is that you can get Improved Two-Weapon Fighting as a bonus feat (ignoring the Dex requirement, even!), and then take Favored Power Attack from Complete Warrior, which at least makes your Power Attack useful against a couple of types of foe, and the Distracting Attack variant from Player’s Handbook II, to help the rogue out against enemies you can’t grapple. You also get a lot of skill points and a couple of spells, to make you less of a one-trick pony. The idea here would be Barbarian 1/Ranger 6/Bear Warrior.
Horizon Walker
Horizon walker is a core prestige class (i.e. technically it’s from Dungeon Master’s Guide not Player’s Handbook), but you can squeeze two levels of it in before qualifying for bear warrior. It requires Endurance and 8 ranks of Knowledge (geography), so it works best with at least three levels of ranger.
The big advantage of horizon walker is the desert terrain mastery. That gets you immunity to Fatigue, which means you can end your rage early without becoming fatigued until the end of the fight. Plus you’re immune to other sources of fatigue and exhaustion results in fatigue, instead, which is nice because fatigue is bad and exhaustion is awful.
For the second level, underground’s 60-ft. darkvision is almost certainly your best choice; a +4 competence bonus to a skill is just not all that exciting.
After you finish bear warrior, returning to horizon walker long enough to get a planar terrain mastery means you can take the shifting mastery, i.e. the ability to use dimension door once every 1d4 rounds. Dimension door is far from a great spell, but it is teleportation which can be difficult to come by for a martial character. The cavernous mastery’s 30-ft. tremorsense isn’t bad either, though by that level you really need to have gotten into the air.
Ideal build is probably just Barbarian 2/Ranger 3/Horizon Walker 2/Bear Warrior 10.
Hexblade
This class from Complete Warrior is not great. Its curse is Charisma-based, which is fairly awkward for you (could try to be intimidating, I guess), and what it does isn’t all that great. But it is full-BAB, has a good Will save which is useful to you, and the arcane resistance and mettle class features are decent. But what really makes this worth even considering is Player’s Handbook II, which offers one variant on it that makes it useful to you: the dark companion.
The dark companion replaces the familiar, and provides a 3×3 square area of −2 to AC and −2 to saves. It’s just an illusion, and it’s painfully vulnerable to dispel magic, but those penalties are pretty significant. It can move independently of you, and has pretty big range.
Effectively, it allows you to serve two roles for your party: locking one enemy down with your grapple, while making another enemy vulnerable to attacks and spells.
That takes four levels, barbarian takes two. The last level should be fighter for feats or ranger for skills.
Ex-Knight
This is a weird option, but I kind of like the story it tells: a knight who lost his civility and nobility, becoming savage and warmongering. Sounds neat, anyway. You start as a knight from Player’s Handbook II, take it for three levels to get bulwark of defense, and then change alignment, forsake knight, and go to barbarian. You lose the knight’s challenge features, but you maintain the Mounted Combat bonus feat (though you’ll probably never use it), shield block +1 (which you’ll definitely never use), and bulwark of defense (which you definitely will use).
Bulwark of defense makes you “sticky”—people next to you will have a hard time becoming not next to you. Less useful since presumably you’ll be grabbing them, but your grappling won’t exactly be stellar so this provides a decent back-up for things you can’t grapple.
Three levels of knight and two levels of barbarian still leaves two levels. Ranger remains probably the best option, just for the skill points (Favored Power Attack maybe if you can pick a foe you really will see all the time), though fighter could work for feats. A fourth level of knight gets you armor mastery (medium), which is OK if you want to go with mithral full-plate as your armor.
To Increase Your Survivability in Combat
The only question to be found above was in your heading, so I'm going off of that: how to get the most AC and HP as a Bear Barbarian. But I'll also focus on making that AC and HP work for you: specifically in helping you take less damage in the first place, and give you some partial healing abilities.
The first advice I'd give for increasing your AC is to raise your Dexterity up to 20 ASAP. This will give you an AC of 10+5+5+2+2=24. (With your Unarmored Defense, and two shields). From there, you might want to see if you can grab some magic items like the Cloak of Protection, or Ring of Protection, both of which add +1 to your AC and saving throws. And naturally, a couple of Magic Shields couldn't hurt, as many of them raise your AC without requiring attunement (credit to daze413 for this point). But some of the best defense bonuses you can get come from multiclassing1.
Multiclassing:
Your Bear Barbarian's best trait to help your survivability is his resistance to all damage, when raging (other than psychic). But raging won't allow you to cast or concentrate on spells. So any further aids to your defense will be based on routes other than spellcasting.
For Being Hit Less/ Taking Less Damage from Hits:
Counterintuitively, you might want to take a several levels in Rogue or Monk. With your AC so high, you're unlikely to get hit by much other than spells. By taking up to level 7 in Monk or Rogue, you'll gain Evasion:
When you are subjected to an effect that allows you to make a Dexterity saving throw to take only half damage, you instead take no damage if you succeed on the saving throw, and only half damage if you fail. PHB p-96
Since Barbarians have advantage on Dexterity saving throws already (Danger Sense), this could add up to considerably reduced damage for you. Not to mention the the "half damage" guaranteed by this ability would stack with your "resistance" causing you to take a quarter damage overall, worst case scenario (if you are raging).
Rogues main combat advantage is their sneak attack, which you couldn't use unless your DM considers shields to be finesse weapons (unlikely). But you'd also get the 5th level Rogue ability of Uncanny Dodge, which would allow you to use your reaction to halve damage from one attack per round (with your Reaction) that manages to hit you (which would mean you'd take a quarter damage overall, since the Rogue's ability is not "resistance", and would stack with your current resistance to damage while Raging).
Monks, on the other hand, gain an ability to dodge as a Bonus Action at level 2, at the cost of a "Ki" point. If you took up to Monk level 7, you could do this seven times per short rest, which would be quite effective at keeping you from being hit. You'd also gain some control and damage ability from the Monk's "Stunning Strike", and healing ability from the Monk's "Wholeness of Body" (if you chose Way of the Open Hand as your Monastic Tradition).
Adding to your HP/ Being a Damage Sponge:
You've requested to "avoid magical classes". I don't know if you mean you want to avoid classes that require casting spells (which makes sense because of your Raging restrictions), or if you were opposed to classes that have spell slots overall. If it's not the latter, a good choice might be a Moon Druid.
Technically, transforming into an animal isn't a "spell", so you'd still be able to maintain a rage in a transformed form, and transform into (and out of) it as bonus actions while continuing to rage. This would give you all the HP of the beast you changed into, which would be effectively doubled since you will still take half damage while you Rage. Also, you'd still be able to use the Druid's spell slots to heal yourself while transformed.
Additionally, while you are transformed by Wild Shape, you can use a bonus action to expend one spell slot to regain 1d8 hit points per level of the spell slot expended. PHB p-66
Since this isn't a spell either, it's compatible with your Raging, allowing you to take half damage and still heal yourself.
The downside to this build is that the transformed shape is unlikely to be very high AC. Even using your "Unarmored Defense", most beasts still would have an AC of 15 or lower. Still, it's one of the highest survivability builds out there, and when the beast HP goes down you're back to your high AC self with full HP.
Personally, I'd recommend Barbarian 5, Monk 8 (ASIs in Dex), Moon Druid 2 for your level 15 character. It would provide a combination of incredible evasion (dodging most rounds) and a major HP buffer, both of which are replenish-able on a short rest. You'd only get 3 Rages per day, but that will allow you quite a bit of time where you are an unhittable, unstoppable steel-skinned damage sponge.
1 Small addendum to multiclassing: Barbarians gain the highest amount of hit points every level per class. So all of this multi-classing will cost you about 20 hit points to your regular HP, since Rogue, Monks, and Druids all earn 2 fewer hit points per level than Barbarians. But between the 74 hit points per short rest you're getting (from transforming into a Dire Wolf or Tiger twice, with 37 HP each), and the extremely increased damage reduction you'll get from Monk (near constant dodging and 0 damage from succeeding on an extremely common saving throw for which you have advantage), I firmly believe you're coming out ahead.
Best Answer
If you can get your DM to allow use of Unearthed Arcana: Class Feature Variants, you can make use of Martial Versatility (replace Fighting Style with another on levelup) and Unarmed Fighting style, which works well with grappling:
So, in the end you would deal d6+d4+str+rage damage to a grappled creature for 5 average damage per hit increase from what you have now.
EDIT: Unarmed Fighting was printed in the Tasha's Cauldron of Everything with one difference: instead of damaging enemy on grapple start and every hit, you instead deal automatic damage to a single creature at the beginning of your turn.