So, you have a couple of options for when you get your visa documents and passport back:
1) Reapply for the fiance visa, selecting the correct visa type this time, then move to the UK, get married, and switch to FLR(M). You won't be able to apply for FLR(M) in person though, since you will have a previous refusal, which makes it a non-straightforward application.... so I would allow a couple of months or so after the wedding to get it by post, and to be able to start studying
2) Apply instead for a Marriage Visitor Visa to marry in the UK, then come back to the US and apply for a spousal visa, but with a previous refusal, it could take a few weeks to be processed, even with priority processing... so you might not get it until Sept/Oct
What are your timings?
What date are you planning to marry?
What date does your PhD start?
Can you push the start date of the PhD back at all?
Alternatively, can you get married any earlier, to give yourself more time to get the next visa?
One other possible option, though not sure if it would be feasible for you - your partner could fly to the US and you could marry there without needing any visa at all (and only 1-2 days waiting time). Then you could just apply directly for a spousal visa (or even a student visa) without worrying about timing. You could even still have the Belfast wedding, but just make it a blessing after the legal wedding, rather than it being the actual legal ceremony.