Barbarians also have something that already covers the bare-chested thing, their unarmored defense. If you want Slayers to go down that path I'd advocate an incentive rather than a restriction. Let them use Str + Con + Dex or add Prof bonus when not using a shield, so they will want to rather than be forced to because that's what the lore says.

You know, you could implement some interaction with the DMG Honor stat. Have Slayers require an Honor stat below say 13 or so, like a reversal of the typical multiclass requirement.

Of the dot points I like the last one best, it suits the flavor and ironically could be a semi-frequent occurrence if the player is on-board. I'd say make it a minute instead of a round to make up for the party losing a member (and all their actions/damage). The best (and I think funniest) part is that Resurrection magic is a thing from level 5 so it's entirely possible to just keep bringing the poor sod back after every death that isn't heroic enough.