i wouldn't. once the bichirs get bigger, they could shove the mesh if it isn't really secure. plus, the betta fry will simply swim through the mesh and then you'd be back to what i mentioned before - the predator eating the fry faster than they can hatch. it would also be very stressful for the male betta caring for the fry while they are still in the bubble nest, and he could be likely to eat them all or abandon them because he is stressed. breeding bettas can be a rather delicate adventure!
best to have a separate tank for the betta breeding to happen in.
i would suggest picking up two used 10 gallons off your local craigslist. you should be able to get them pretty cheap. one divided for the males, and one for the breeding tank.
ah, one thing i just thought of - if you are merely breeding fish to feed the bichirs, why not go with something that is much easier to breed, is more prolific, and you only need one tank for? something like guppies? much easier IMO!