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.
Its viability depends on external factors
It is mostly viable, actually the only viable ranged Barbarian build in my opinion1.
The external factors are:
- party composition
- available space
- encounter type (if it is a single boss, or many smaller monsters)
- enemy type (if it uses attacks)
When I played this build, it was quite useful when everything was optimal.
However, its usability decreased significantly with one of these factors missing.
Party composition
As mentioned in the comments of this answer, this tactic is only good if your party members can keep the target of your Ancestral Protectors (Target) away from you.
Either they have to physically hinder its movement, or discourage it with threats of Opportunity Attacks.
You will want to have at least half of the party members to be in melee.
When the Shield Master fighter was not present at our encounters, my Barbarian worked a lot worse.
Available space
If you fight in a dungeon, sometimes you just can't get far enough from the target while maintaining line of sight.
Encounter type
If there are lots of enemies, you might get far enough away from the Target, but his comrades still might be adjacent to you, causing disadvantage on your attacks. You have to hit your target to trigger Ancestral Protectors, so this can be a big problem for you.
Enemy type
If the enemy does not use attacks, only breath weapons or spells, Ancestral Protectors become irrelevant. Unfortunately the resistance is only against his attacks.
Conclusion
It is not viable for dungeon crawling with a party of 4 Sorcerers, but with the average party in the average encounter you will do more than fine.
Recommendation
Use a Hand Crossbow instead of a Longbow and multiclass to Rogue instead of Fighter.
Hand Crossbow
With the Crossbow Expert feat you can attack without disadvantage even if you are adjacent to enemies, negating the worst consequences of a tight space and many smaller enemies.
When you already have the feat, Hand Crossbows give you the most DPR .
Rogue
2 levels of Rogue make Disengage a bonus action, very valuable in tight spots.
Also a Rogue Multiclass provides better DPR increase than a Fighter2, and this area is where you are very much behind a usual Barbarian, as you can't use Reckless Attack and the damage increase during the rage.
1 Being at range gives you extra survivability, by making it harder for others to target and hit you. Ranged characters come away from most fights undamaged in experience, so resistance is mostly wasted on them, and no other significant Barbarian features are usable at range. Other ranged Barbarian's usefullness in combat is below that of a Ranger, and they are famously weak.
2 A single level of Fighter might still be a good idea for Archery fighting style.
Best Answer
You should be able to play this
It is in no way optimal, but it's certainly possible to play this kind of character, and it will provide you with great roleplaying opportunities. A crippled fighter who uses what little Druidic magic he knows to get by where his physical form can no longer help is a great concept.
As you've already determined, having a low strength and a low dex, you are stuck with having terrible AC unless you wear heavy armor, which has a strength requirement unless it's a mithral armor. You will be able to use a shield alongside a club (heavy stick) for a bonus +2 AC, which means you'll be a fairly tanky character if you don't dump-stat your con as well for the theme.
You will also need to use your bonus action every minute to cast Shillelagh or you'll struggle to hit anything, but once you've done that, you're essentially just using a one-handed longsword, which is a perfectly viable build. You'll be weak to dispell magic and anti-magic zones, but in all honesty, if somebody is going to cast that to dispell your cantrip, they've wasted a spellslot.
For your once per day spell from Magic Initiate, you might consider Absorb Elements. Because you'll be dump-statting your dexterity, there's a decent chance you're going to fail dexterity saves as you are not proficient in those, and a well-timed Absorb Elements might well save you 10+ damage from a spell.
For your other cantrip options besides Shillelagh, you might want some way to 'close distance' or attack from a range, for those situations where your movement (especially when you're slowed down by armor) can't get you near enough to the enemy. With a decent wisdom modifier, your Thorn Whip would allow you pull enemies closer reasonably well. Gust might be useful to push enemies over edges from a distance or push unattended weapons around, allowing you to be more useful from a range than your slow speed would normally allow.
So in conclusion, is this going to be the strongest character at your table? Almost certainly not.
Is it going to be unplayable? I don't think so. I think that once you find Mithral Armor or find some other way to get a decent AC, your character will be perfectly viable in anything but a heavily min/maxed environment.