I've recently become intimately familiar with the GOES website enrollment process, first for a NEXUS card and then for the Global Entry program. In my case there was a glitch, not related to any previous criminal conviction but rather to the fact that I hold two passports, and I made the mistake when I enrolled of listing my US passport as secondary (using my Canadian passport as primary, since that's where I currently reside).
The result was that I was approved for both NEXUS and the Global Entry program but I have not yet been able to successfully use the kiosks, as the Global Entry program does not recognize me as a US citizen. Clearly, just a software glitch, albeit an annoying one. (Despite this, NEXUS entry works flawlessly at YYZ in both directions.)
The point of all this is to say that I have visited Global Entry program offices in MIA, ORD & SEA in recent weeks trying to sort this out. It may or may not be resolved now (I won't know until I try to enter at MIA next week) but in all three cases the people I spoke with were accessible, knowledgeable and helpful. My recommendation to the OP would be to visit one of the offices to ask how this is handled. They're obviously going to couch any opinions proferred very carefully, but I'd guess you have a decent chance of getting a reasonably direct answer.