As we know Kid losing his arm post time skip remains a mystery. I think I have a theory on how he may have lost it.
We know Kid early on attacked Big Mom’s ships and later actually came into Big Mom’s territory only to be forcefully removed. So much so he changed his target to Shanks. How did he manage to leave without being killed.
He was forced to spin the roulette wheel landing on the arm symbol. Probably a lucky roll when considering the alternatives.
*Theory by themitchellbrothers