I also see Dr. Levine, and coincidentally I have also received an FMT, though not for ME/CFS - rather to treat a severe, recurrent C. Diff infection. For that purpose, it was extremely effective. I did not see any improvement in my CFS symptoms after the FMT, though it's possible I may have, were my body not totally wrecked by the C. Diff inflammation, and by not having eaten any food for nearly a month.
I do strongly suspect that GI disturbances are a key element in ME/CFS pathogenesis for many (though probably not all) sufferers. I notice that my GI symptoms, which are currently something you could label as IBS, are strongly correlated with my CFS symptoms. And this is not merely that my GI symptoms are a downstream effect of my CFS - my CFS symptoms worsen if I eat something that cause indigestion, for instance. And strangely, it often occurs that my CFS symptoms abruptly improve or worsen immediately prior to or after a bowel movement.
I would recommend not getting an FMT for the time being, even if it holds a lot of promise as a treatment. When it comes down to it, there is no such thing as a reputable clinic or donor when it comes to finding a 'perfect' donor. Even when I received an FMT for my C. Diff, working through the most official channels possible, the screening criteria were extremely lax compared to what I think would be warranted for identifying 'perfect' donors. And of course, we don't really even understand what makes a donor perfect - once there is some real, objective benchmark for donor quality then maybe it might make sense to do FMT for ME/CFS.
In the meantime, it might be worth looking into probiotics. I've had some limited success with 'Innovix Mood Probiotics', which contains two strains that have been shown in one study to modulate the HPA axis, by sensitizing glucocorticoid receptors. The impact of gut flora on systemic endocrine function is, I think, sorely underappreciated. Probiotics are a much safer way to experiment with this theory without the high cost, inconvenience, and potentially drastic adverse effects of an FMT.