Is it possible to compete in professional boxing while on active duty?

Ultimately the answer is up to your commander because it is a high risk event since you could become injured and non deployable.
When I was in 4/25 you had to go to the combatives gym and prove your skills and the gym would talk with your commander before a decision was made.

Wasn't David Robinson a lieutenant in the Navy while playing profession basketball? I'm pretty sure its possible under the right circumstances. It certainly would bring a lot of positive advertising to the service -- as long as you're winning.

You can if you are good at it. The WCAP located at Fort Carson deals with that. https://www.armywcap.com/. I had a Soldier who came from that to our unit and he was a third degree black belt who almost made the Olympic team. If you do make it for boxing that becomes your job for the time you are with them.