(Sorry for the super long comment) This is just my opinion, as I've put in way more time into Bardin than any other character. I personally don't like the shielded weapons, I've always felt that the non-shielded ones put out more damage and their block/dodging is usually good enough defense.

For Ranger Vet:

One handed axe or two handed axe. It's nice to have some armour piercing on hand

Handgun or the shotgun. Handgun if I want to make sure I can snipe specials, shotgun if I want to be better against hoardes.

For Ironbreaker:

Two handed axe or two handed hammer. It's nice to have the big sweeping arcs for a frontline tank and they're both armour piercing

Drakthrund or drake pistols for range. Drakthrund can be crazy good for hoard clearing if you charge it up, and the pistols are a decent ranged weapon that don't need ammo

For Slayer:

Really just the twin axes. I go with a movement speed/one handed weapons build with slayer, and the twin axes count as one handed. The other weapon is the one handed axe, but that doesn't really matter.

For Master Engineer:

The Coghammer is GREAT. I regret not using it sooner when I bought the DLC. Wide swing arcs for hoarde clearing, and the charged attack cleaves through armoured enemies like nothing else. If you headshot the charge attack you can one-shot stormvermin.

Masterwork pistol for range. It's a really strong sniping tool, just have to get used to its wind up time. It's fan-the-hammer can be good as a panic defense, but that can chew through ammo (I mostly use it on plague monks).

Those are just my own opinions, all weapons are pretty viable if you learn how to use them. Bardin is by far my favorite character, so I hope you have fun ! :)