Advantages: the fusion side doesn't have to quite hit breakeven, and on the fission side you can handily break apart any transuranic atom, even more effectively than fast reactors. It'll destroy any transuranic waste (which is most of our nuclear waste, and almost all the long-term waste), or directly fission thorium or unenriched uranium.
On the other hand, fission people don't like the complexity and fusion people don't like introducing all that hassle with fission products and decay heat. But it may well find a niche for nuclear waste elimination at least, and produce a lot of energy along the way.
more seriously, it could split the energy production to some ratio of fusion/fission and produce less hard to handle waste. Since the fusion reaction provides the neutron source, it may not be as necessary to provide as fissionable material which may not be able to meltdown?
I dunno, I have no knowledge or experience at all, this is my "if I was a sci-fi writer this is a thing I would put into a story"
But doesn't nuclear fission work off of neutron bombardment? Why not build a hybrid reactor where the "shield" undergoes fission while the core is generating fusion and perhaps they can produce a useful amount of power, perhaps with the fission "shielding" powering the magnets or some such?