The other races don't do +15 extra unarmed damage like Khajiits though.
The point I am of course making, is that the extra buff khajiit get from their claws potantially becomes insignificant in the face of what can be attained through enchantments on armour. Simply get the heavy armour perk, wear daedric gauntlets to get an extra 18 damage, and then enchant all your armour with H2H boosting enchantments and the extra points of damage a khajiit gets is largely irrelevant.
also, Khajiit shouln't be punching if they're supposed to get a bonus from claws, and their bonus shouldn't apply in a brawl where the other guy specifically says 'no weapons'.
EDIT: and @above, the strength of the H2H enchantment is based on your personal enchanting skill, the soul gem used will have a lot less significance with armour enchantments of that type. Set your enchanting skill to 100 through the console and you'll see what I mean.