I'm not sure how much you know about bloodfin tetras, but here goes...
Bloodfin tetras should be kept in schools of 4+ and in a tank of 10 gals or more. They are recommended for the following tank mates: corydoras, small loricarids, other like-size tetras, and hatchetfish. They also like dark areas.
That being said, it is possible that your tetra may be nipping the betta. Consider this, since they're a schooling fish and you only have one and since the enclosed area is small, they may feel insecure. Not to mention that the fish may feel a need to defend itself from the betta... who knows.
You may want to place the tetra in another tank. If infections occur in the betta's fins, finrot may soon follow.
HTH