Can I setup my road bike, mountain bike, and fat bike with the same set of shoes with MTB SPD cleats?
You don't say how long you spend on each bike, and it makes a difference. Many people only spend long enough on one of their bikes for Q factor to be an issue. The other issue is how sensitive you are to fit. 

Personal experience: I've got a tourer and a hybrid (MTB BB, noticeably wider) with SPDs (and sometimes on the hardtail which is also wider than the tourer).  I use the same shoes on all of them. Switching from one to the other, the clipping in takes a little getting used to, which indicates the change. I haven't had knee trouble from switching (though I have had a little trouble recently which I blame on finding my saddle lower than it should have been just after a 330km overnighter). But I tend to only go up to about 70km on the MTB (road-riding to the trails for example) and these days do less than that on the hybrid (though it's fine up to 70km as well). Generally I seem to be fairly easy to fit once the bike is big enough.

Realistically the distance between your feet will change by (very nearly) that much whether you're clipped in or not. All the cleats will do is prevent you compensating by angling your feet. That may even be a good thing.