Re: So here's a thought:
"So, since they've seen emissions for longer than they expected, one possibility is that the central object is indeed a neutron star and not a black hole, and this would be interesting because it would tell us things about neutron stars I think."
More like 'fusion' then - and being inelastic, kinetic energy has to go someplace... and so it spins so fast that it must slow down somehow, so emits gamma [and maybe absorbs a good deal of that, too, like you suggested]. But when the neutron star absorbs it, what does it cause to happen? I was assuming that it's behaving like an excited atom, which must either emit gamma or fission to remain stable. On a macro scale.