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.

Take the Mobile feat

Your goal should be to get into melee, apply your ancestors to the boss, and get away. With the Mobile feat (PHB, p. 168), you won't provoke opportunity attacks from enemies you attack, and you'll have extra movement (in addition to a barbarian's own extra movement) to leave range. If at least one of your allies stays near the enemy, the enemy has a tough decision to make: eat an opportunity attack (or three) or attack your now very tanky allies.

In practice

One of my current Adventurers League characters is an Ancestral Guardian / Battle Master using this strategy. When I originally wrote this answer, he was an Ancestral Guardian 6 / Fighter 2. Since then, he has capped out at Ancestral Guardian 16 / Battle Master 4.

Throughout all tiers of play, he has been extremely effective at preventing damage on myself and my allies. Why be a Bear Totem barbarian when your ancestors turn your whole party into Bear Totem barbarians! As far as encounters and magic items go, AL tends to have 3-4 encounters per module, and my character only had a +1 weapon for levels 5-7 without feeling weak.

Other considerations

• You mention that the Champion fighter in your group is cowardly. The nice thing about this build is that (unlike the Cavalier from Xanathar's) you can...encourage...allies to synergize with you without forcing them. Fighter not tanking? Apply your ancestors and use your 50' of movement to run past your cowardly ally. The enemy will likely engage with the fighter since it can't reach you. When it misses the fighter, play up how your ancestors distracted the enemy. Your allies will learn that ancestors equal safety.
• It's usually better to apply your ancestors to the least damaged enemy first and then leave them so you can attack the most damaged enemy with the rest of your party. Your ancestors do nothing if the enemy is dead.
• If an enemy is highly mobile and dangerous, you can stand back and apply your ancestors from 150' away with a longbow.
• If you also multiclass into fighter, the Battle Master's Menacing and Goading Strikes (PHB, p. 73-74) are like a mini ancestors effect. They synergize well with the rest of the build.
• As @Ben Barden pointed out, the Sentinel feat on the fighter (PHB, p. 169-170) in your party would assist with this playstyle by preventing even mobile enemies like dragons from leaving to hit you.
• I would not recommend taking any feat or other feature that uses your reaction. At Level 6, you get Spirit Shield, which prevents an average of 7 damage per round as a reaction. That's similar to casting healing word every round without using resources! With the hit and run playstyle, you often won't be next to enemies to use an opportunity attack, so you can (and should) use Spirit Shield every round.

Bonus math

This build always feels strong in play, but I was wondering how strong it actually is.

TL;DR: the wizard, protected by ancestors, survives longer than the reckless-attacking barbarian.

The PCs

Since this question focuses on survivability, I am assuming that the barbarian is using a one-handed weapon and a shield. If not, a hit-and-run playstyle becomes even better. (I would recommend a two-handed weapon to anyone trying this build.)

The monsters

This fight will be a Deadly encounter, as it is the boss fight. The PCs will face:

• One CR 5 Gladiator (MM, p. 346)
• Five CR 1 Duergar (MM, p. 122)

A fireball from the evocation wizard will seriously injure or kill several Duergar, leaving the rest for cleanup by the warlock and the ranger/rogue. With that in mind, let's focus on the Gladiator. The Gladiator attacks three times with a +7 to hit, each dealing 11 damage on average (18 on a crit).

The ancestors

I will assume that the barbarian's ancestors are applied to the Gladiator every round. (There is a small chance that they will miss both attacks, even if they attack recklessly for advantage.) For anyone but the barbarian, the ancestors impose disadvantage on the Gladiator's attacks and will provide resistance against any damage the Gladiator may do.

The numbers

As a baseline, the barbarian will last 4.4 rounds against the Gladiator while attacking recklessly, or 6.9 rounds normally. The barbarian numbers assume that they have resistance to the attacker's damage. If the party faced a CR 5 Fire Elemental instead of the Gladiator, the barbarian no longer has resistance, but their ancestors still protect the party against all attack damage.

The wizard can last around 5 rounds protected by ancestors if they cast shield on the first two turns of the fight. If the barbarian took the Tough feat (gaining +2HP per level) instead of Mobile, they would last 5.2 rounds recklessly, or 8.2 rounds normally. These are still a far cry from the 18.4 rounds the fighter would survive.

The conclusion

The fighter becomes a truly incredible tank, surviving more than 5 times longer than without the ancestors. In fact, every PC survives longer than a barbarian who is attacking recklessly (all while allowing the barbarian to do so without penalty by keeping out of range). Who knows, you might even make a brave warrior out of your cowardly fighter ally!