How would I go about creating a mod that adds bruises or cuts onto sims after a fight? Also, how would I make this compatible with a mod like adeepindigos Healthcare Redux (have fights potentially give sims temporary injuries).

The advanced XML modder would override the vanilla fight interaction with a custom one which adds some tattoos (injuries) to the sims. Sooner or later this mod breaks, so it is not a good idea to implement it like this.

A script mod is usually more stable when altering existing interactions.
I'd patch the fight interactions during startup and append a basic_extras/do_command to it so I can attach CAS parts / tattoos via script and vary the results.
Or you might inject into the interaction queue and listen for fight interactions to do the same.

The other mod should use the same tattoos or skin overlays and be able to remove them. They might even support traits/buffs for sims with injuries which they remove so you could also set them.