Ok, so I started a new character who had a very high unarmed skill.

I decided to go down to New Reno as quickly as possible to become a boxer.

I won every fight by a landslide until I reach the guy who bites peoples' ears off, whenever I try to fight him it instantly ends the match and says he won.

I know this has to be a bug but is there any way to get around it?

You could try installing the unofficial patch (which we probably should have recommended when you were having install issues, oops), but you'd have to start a new character since Fallout and Fallout 2 can't be modded mid-game like Fallout 3/NV can.

Depending on your progress it may be more practical to simply forget about finishing the boxing ring in this particular play through; unless someone else has another solution.