No meat does not rot in your gut.
The meat takes a bit longer to digest than other foods in your gut and it usually stays longer in your gut than other foods but the meat is digested and pooped out before the meat does actually rot.
Usually the meat is digested and processed through your digestive tract within a few days but the meat should be digested in your stomach within 24 hours fully.
Then it can take a few days to have it go through your gut and intestines and be pooped out.
Whenever I eat meat such as steak or hamburger I usually do not need to poop until a few days later but with some other foods such as bananas I need to poop the same day.
The meat though or any other foods will not rot in your guts or intestines.