[RPG] As a party meat shield, should I start with levels in Barbarian, or Druid

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.

As is the usual advice, I would say that talking to your DM and talking to your fellow players is a good place to start. Express your concern to your DM...he's the one who can help you with this, not us.

But, I don't think it's necessary just yet. You still have a significant role in the party.

A few points to consider...

1. The Bard was almost certainly going to surpass you in Intimidate if he took Proficiency in that skill. Bards are a Cha-primary class who, at high level, get access to the spell Glibness. Only they and Warlocks can cast that, and you simply can't compete with a guaranteed 15 or better on any Charisma roll. I mean...that's a major feature of Bards. Insane Social Skills. This doesn't mean you can't Help though. Be big, threatening, and intimidating. Loom behind the Bard's shoulder while he does the talking. Crack your knuckles or rub the haft of your axe at meaningful times...give that Bard Advantage on his check.
2. In terms of damage...yeah, Rogues do stupid amounts of single-target damage. That's kind of their whole schtick. But, in exchange, they are relatively fragile. Against a single target smacking them once a round, they are fairly good at not dying. But they'll go down like a chump against a mob of enemies. You won't. It's kind of a trade off. I mean...an angry 20th level Wizard can take out a significant portion of a city in a few rounds...but they are also made of glass. You don't hit quite as hard as the Rogue, but you hit hard enough to meaningfully contribute to the party's damage output, and you can get hit in the face a LOT MORE without going down.
3. You are still significantly better at Strength-related activity than the Rogue, unless he has Expertise in Athletics. As a Barbarian, you have Advantage on all Strength Checks that you make while Raging and, because of your 15th Level Persistent Rage, coupled with the Unlimited Rages of 20th level...you should be making every strength check with Advantage, simply by firing up your Rage before you attempt it. If you want to see the actual math, you can see it in this AnyDice Program. In it, I compare the Rogue's raw Strength Check with your Advantaged Strength Check (both at a +7 to their Strength Check). Their average roll is a 17.5, yours is 20.8. And with a narrower Standard Deviation, you will consistently churn out better results than the Rogue. (And if you took the 6th level Bear feature, you can carry/lift twice as much as they can without having to make a check at all)
4. The Rogue should probably always have been the infiltrator all along, even if he wasn't as good at climbing walls. Not being noticed is generally more important than getting over the walls in one attempt.
5. You are still, quite definitively, the party tank. You almost certainly have the most hit points, and since you should be perpetually raging while in combat, you have resistance to everything but Psychic Damage. And the 14th level Bear trait, coupled with Sentinel makes you fantastic at keeping enemies focused on you...so the comparatively fragile Rogue can hit them like a freight train. In the entire party...you're the one who can take a Dragon's breath weapon to the face and shrug it off without flinching.

Ultimately, your choices while leveling your character resulted in a character that is immensely hard to kill, and can probably throw a horse at someone they don't like, and rip the portcullis off a castle. And he's really good at keeping enemies focused on him. You're good at being scary....but being big and threatening only goes so far compared to the clever words of a Bard who can make all sorts of insinuations and suggestions that are far scarier than a big muscular guy. Or perhaps he just knows the right things to say to make you seem even scarier.

You have somewhat pigeonholed your character into this by your Feat selection, class build selection, and ASI choices and, truthfully, barbarians have a fairly narrow class focus anyway. A Bear Totem Barbarian's primary focus is "I'll keep their attention, they can't really kill me, you guys pummel them." As D&D is first and foremost a game focused on combat...the ability to keep enemy attention focused on the guy who is near-impossible to kill is a very valuable resource to bring to the team.

Seriously...just see how your fellow players feel about the prospect of going into combat without their favorite meat shield. Based on my prior experience with tanky characters....they'll think about all the damage you soaked for them...consider all that damage hitting their characters, and not like the idea one bit.