An analogy: http://eccc.hpi-web.de/report/2011/108/

I'm afraid I don't see how this article is analogous. The article points out that computational complexity puts a very real limit on what can be computed in practice. Thus, even if you'd proved that something is computable in principle, it may not be computable in our current Universe, with its limited lifespan. You can apply computational complexity to practical problems (f.ex., devising an optimal route for inspecting naval buoys) as well as to theoretical ones (f.ex., discarding the hypothesis that the human brain is a giant lookup table). But these are still engineering and scientific concerns, not philosophical ones.

Need it be primarily one or the other? But if I must pick one, I pick philosophy.

I still don't understand why. If you want to know the probability of FAI being feasible at all, you're asking a scientific question; in order to answer it, you'll need to formulate a hypothesis or two, gather evidence, employ Bayesian reasoning to compute the probability of your hypothesis being true, etc. If, on the other hand, you are trying to actually build an FAI, then you are solving a specific engineering problem; of course, determining whether FAI is feasible or not would be a great first step.

So, I can see how you'd apply science or engineering to the problem, but I don't see how you'd apply philosophy.