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
Polearm Master should be usually better, but Spear Mastery has the potential to deal more damage on early levels
Both weapons will deal 1d10 damage on hit. Spear Mastery gives +1 on the modifier and allows you to use your Bonus Action + Reaction to attack someone who enters your reach, for an extra 1d10 damage.
So, let us consider a first round, with and without Rage (I will use +2 as the rage damage). For now, let us ignore the +1 bonus on the modifier from Spear Mastery.
PAM
On hit, you will deal an average (5.5 + 3 + [2]) on the weapon hit, then (2.5 + 3 + [2]) on the bonus action hit then (5.5 + 3 + [2]) on the reaction, for a total of 22.5/[28.5] (without/[with] rage).
Spear Mastery
On hit, you will deal an average (5.5 + 3 + [2]) on the weapon hit, and (5.5 + 5.5 + 3 + [2]) on the reaction, for a total of 22.5/[26.5] damage.
Surely, with the +1 to hit, the Spear Mastery will outdamage the PAM without rage, and, depending on the enemy's AC, may outdamage with rage as well.
But the thing is: most combats do not involve enemies constantly leaving and entering your range. With PAM, it is fairly simple to move one step back, and keep this type of control. On the other hand, Spear Mastery requires you to be at least 20 ft. away from the enemy when you use the bonus action. I have been playing a PAM Fighter and quite often I can not use my reaction, because enemies are not that dumb, and will find a way to dodge me, attack from range or whatever tactics that nullifies my reaction attack. As an example, many times, combats happen in dungeons where your movement is severely limited, and constantly kiting and being 20 ft. away will be a problem.
Finally, Spear Mastery is situational in the sense you need to call one enemy to target. If that one enemy does not enter your range, but all others do, you can not attack them, and your bonus action is lost.
So, when we consider the scenario that most likely is going to appear on a second round of the combat, with enemies already in range, PAM will deal two attacks, and Spear Mastery will only be a +1 bonus (although it does free your bonus action). In this case, PAM will always outdamage Spear Mastery.
There is also the problem of actually activating Rage: it costs a bonus action. With PAM, you are losing the 1d4 attack, but with spear mastery, you now can not use the reaction from the feat any more, as it requires both the bonus action and the reaction.
And, finally, PAM will get even better as you level. For every extra point in Strength (modifier) you get, the PAM will increase 1 more damage on hit compared to Spear Mastery, and the same for Rage Damage, same for Magic Weapons, etc., as PAM allows one more attack than SM.
For comparison, consider a 9th level Barbarian, with 20 Str and +3 Rage damage. SM is slightly better for ACs 22 or higher, which are not that common, and PAM is slightly better elsewhere, considerably better for low AC enemies (auto-hit enemies, AC <= 11). The figure is done considering you can actually use all your attacks.
Then again, consider how more situational the Spear Mastery is, and PAM should be the superior choice.
So, to conclude, Polearm Master is considerably less situational than Spear Mastery, allowing it to be used more frequently, and, as you level, it will become superior in every way. The only scenario I can see Spear Mastery being more useful is if you think you are going to face many enemies outside your range (and that you can not approach), in a way that constantly throwing your spear is something that will show up often. In my experience, this setup is very rare, so I would go with PAM.