What if, when you would normally hit the damage bonus cap, monster-family damage bonuses still boosted your damage against monsters?
Say you wear the Black Kat set, Barbarous Thorn Shield, and wield a Wild Hunting Blade. Your damage bonuses with guns, bombs, and swords against most enemies would still be capped at +6, but against beasts you would have an effective +9 when using the Wild Hunting Blade.
Weapons with inherent monster damage bonuses would still be inferior in most cases, but at least they would have a niche and crafting them wouldn't be considered such a terrible mistake. Monster damage Unique Variants would be viable, and perhaps even sought after. While not an ideal solution, the extra potential damage could at least partially address the buffed HP of Shadow Lair monsters.
It's a popular suggestion, but I think it would ruin the game.
It would force people to get +4 monster damage bonus UVs on their weapons if they want to be competitive, because getting to +10 damage bonus is crazy strong. I am assuming there'd also be a +6 cap for monster bonuses? If not you'd probably want to wear something like Shadowsun Stetson, Deadshot Mantle and +4 damage from trinkets +3 undead perk and a +4 undead UV to get a +15 damage against undead on all your guns. If there is a +6 cap, then you could still get to +12 against undead. In any case, damage bonus like that will make your net damage output much higher than someone who sticks to the old +6 universal damage bonus. Damage bonus is just really powerful and that is why people insist on running +6 damage in the current meta. That is also why there is a cap in the first place.
I think monster damage bonus UVs can already aid players in the new Shadow Lairs. If you have +4 against a monster family that is weak to the weapons damage, you can suddenly use it with a more defensive armor set rather than being forced to use Black Kat. Yes they are not as useful when you want to do all levels with the same loadout, but for specific levels that +4 against X is going to be just as effective as a +4 against everything.
Making their damage bypass the cap is going to make the game super pay 2 win, because you'll need to roll +4 damage UVs on all your weapons, preferably in addition to ASI or CTR VH UVs if you want your gear to be "perfect".