On speccing against undead

Without employing too much meta knowledge about the specific monsters of the adventure, we can use a bit of knowledge about undead to make decisions (which is probably justifiable as in-world knowledge too).

Your thought about bludgeoning damage isn't bad, but it's only skeletons which have a vulnerability to bludgeoning damage. A few undead (such as Shadows) have vulnerability to radiant damage, so grabbing that would be valuable. Doubly so, since very few undead are resistant (or immune) to radiant damage, so having it would be useful to have some consistent damage in the face of resistances to non-magical attacks. (On that note, you'll probably want to get a silvered weapon as soon as you can afford it, if not supplied with a magic weapon by then.) Conversely, necrotic and poison damage should be avoided in particular.

Barbarian zealotry

Accepting the above as showing that we want to lock into bludgeoning+radiant as our damage suite, we can start choosing things for our Cleric Barbarian. To get radiant damage, Path of the Zealot has Divine Fury:

While you’re raging, the first creature you hit on each of your turns with a weapon attack takes extra damage equal to 1d6 + half your barbarian level. The extra damage is necrotic or radiant; you choose the type of damage when you gain this feature.

Path of the Zealot is also one of the few barbarian subclasses that add damage, and quickly the most linear for that. As Groody's answer mentions, Totem (Bear) would give you a lot more resilience, though with zealot you could rely on being cheap to revivify.

You're pretty open as to race, though Aasimar will get you some additional radiant damage, Orc or half-orc would offer some more damage on crits (see the notes for Brutal Critical), and variant human will be an option for picking up feats earlier.

Since polearms don't really come with bludgeoning damage (outside of the butt-end attack of Polearm Master) and the classic Greataxe even less so, we can look at two primary weapon choices: Maul and two-weapon fighting.

Modelling Damage

Rather than doing a full damage analysis, we can do a couple of simplifications and assumptions. Firstly, we'll only compare damage during rages. The accuracy of that will depend on your table and/or campaign, so it's not possible to get around it without some kind of assumption. Second, we assume a representative hit chance of 65%. The DPR can then be calculated as: $$ \text{DPR} = \sum_\text{attacks}\big[h (D+m) + cD\big] $$ where h is the hit chance, D is the die damage, m is the static modifier, and c is the crit chance (0.05 for normal attacks, 0.0975 with advantage).

For simplicity, I'll be dealing with a Strength mod of +4, and rage bonus of +2, and I'm ignoring Brutal Critical, and the Divine Fury damage. I'll get back to how these shift things afterwards. The relevant feats considered here are Great Weapon Master for the maul (ignoring the bonus action attacks) and Dual Wielder for the two-weapon mode.

As mentioned, there are a few things not considered in the above analysis.

• Your strength modifier won't actually make a difference between maul and two-weapon, since with the Two-Weapon Fighting style it won't be added to the bonus action attacks.
• Your rage bonus increasing (as it does at 9th) favours Two-Weapon in a way not shown here, since you do get to add that to the bonus action attack.
• Similarly, Brutal Critical also slightly favours two-weapon since it'll only add 1d6 to the critical hits of maul attacks, whereas with two-weapons it'll affect the "whole" damage die.
• Divine Fury is also slight more consistent with two-weapon fighting, since you'll get one more opportunity to hit, and it only cares about the first hit each turn.
• IIRC, a lot of undead have relatively low AC for their CR, and you may well be fighting higher counts of lower CR enemies which would give you a higher hit chance than the 65% assumed here. That would heavily favour any usage of Great Weapon Master.
• Two-Weapon fighting consumes a lot more bonus actions than using a maul. That's relevant for your first turn, since Rage also uses a bonus action to activate. If you end up with other bonus action features, those will also eat into your damage while benefitting from the almost obscene synergy between Great Weapon Master and Reckless Attack.

Other than the feats, you don't actually have any major investment into one or the other, so you may well want to consider using two-weapon fighting during early levels and/or against high AC opponents, and switch to the Maul once you have Extra Attack and Great Weapon Master.

For more on two-weapon versus heavy weapon barbarians, see: How does the barbarian's bonus damage from Rage interact with two-weapon fighting?, Is dual-wielding superior to great weapons for a raging Barbarian?, Great Weapon Master vs. Dual Wielder for a half-orc Barbarian

Running the same analysis on Polearms we get:

Obviously, this means forgoing most of the bludgeoning damage (the relevance of which depends on how anti-skeletons you are) and it needs two feats locking you into v.human if you want to pump your Strength up to +5. You otherwise get kinda the best of both worlds when it comes to the damage discussed above; getting two-weapon fighting's efficiency on Divine fury and Brutal Critical.

If your group has a player planning on playing a Cleric, I would check in with them to make sure your character doesn't step on their turf thematically.

This Is Weak and Limiting

This Way essentially forces a Gunslinger into taking Munitions Crafter. I'll rate the features on a scale of -5 to +5, where -5 means "significantly underpowered compared to the Way of the Drifter", +5 means "significantly overpowered compared to the Way of the Drifter", and 0 means "as balanced as Way of the Drifter". We're hoping to sum up to 0 without any major swings (e.g. a +5 is always bad, even if the total is low).

Reloading Huck

At low levels, you'll only be able to use this for some of the day - those levels are where your bombs are comparatively the strongest, though. So where a normal Drifter might be able to make 10 strikes in a day with their Reloading Strike, you don't have the ability to match that until level 5. There will be an awkward dip in effectiveness at levels 4 and 5 until you get Munitions Machinist which then causes your bombs to somewhat keep up with the melee damage you'd expect from a Drifter. With splash damage and the ability to do this at range (and some good bomb-rider effects), the new Reloading Huck should actually be stronger than it's equivalent. But there's one huge difference - you have to keep interacting to draw your bombs. Neither Quick Bomber nor Quick Draw will help you here, since those are their own actions and thus not usable with Reloading Huck. Given that Quick Bomber is essential to any bomber style of play, this basically prevents the Way from being effective right off the start. Into the Fray will let you start with a bomb, which is great. But the action tax here is huge, so I'll rate this as a -2, for the feat tax and action tax - you've essentially (accidentally) re-introduced the reloading problem that Slinger's Reloads are designed to solve in the first place (that reload is boring and expensive).

Into the Mob

This is fine, it's basically the same, and you still want to be close to enemies for your bombs. +0

Finish the Lob

Tempted to give it a +5 for the punny name alone... but, similar to Reloading Huck, this is crippled by the action tax to continually drawing bombs, and by having bombs left. I'll give it the same rating - -2.

Anarchist's Wake

This is tough; you're probably best off just rewriting this ability altogether. The problem with this is that unless you have a Capacity weapon, you won't have the ability to make three strikes! You'll attack once with your gun, once with your bomb, and then... have to make an unarmed strike? If this build has to invest in unarmed strikes as well, that is another large feat (and item) tax. Not to mention that I'd recommend removing all the unarmed strike clauses anyway as they don't match thematically. The moving without provoking reactions part is also less helpful, since you don't really want or need to be in melee range. So I'll give this a -3 - I think this won't even see use by most Way of the Anarchist Gunslingers.

Anarchist's Juke

If you're not in melee, then Stepping is going to be less valuable. Still helpful, though. The big thing here is that this conflicts with Munitions Machinist, also a level 6 feat. This then becomes a level 8 feat at the earliest (since it's lower priority). So -2 - one because of the feat conflict, another because of Stepping is more valuable in melee. I could also count the action tax of drawing a bomb here, but it's been adequately accounted for elsewhere, and this is an optional feat. I wouldn't take it.

Disruptive Blur

This is fine. Not provoking reactions is more beneficial in melee, but since we'll be drawing a lot of bombs, we'll be provoking more reactions, and probably want to keep at a good distance more. So +0.

Conclusion

Total: -9 - a lot of these features don't work well with the drawing a bomb action tax or the transition to ranged, consumable based damage.

There are a lot of feat taxes for this build, but the biggest problem is it basically reintroduces reloading action tax to the Gunslinger. This makes it weak and limiting. Unfortunately, I don't think it's as simple as letting them always draw bombs for free. However, feel free to try playing it and report the results; bombs are incredibly versatile and that's hard to account for in theory-crafting, so maybe that will make up for it.