Congrats on your wedding and progress in general!
Is there any way to get the PMI dropped without having an appraisal?
It's up to your bank. Hopefully you can work something out. It might not have to be an actual appraisal; they might accept an independent broker's opinion, for example.
I have a hard time believing that an appraisal can't be done. There has to be a way to get an appraisal without comps. Maybe an appraisal for PMI cancellation might be easier to get than one for refinancing, or maybe the bank will let you choose the appraiser, meaning you can call around until you find someone who will do it.
One thing: IANAL, but my reading of the Homeowners Protection Act of 1998 is that if your property value has dropped from the original value, then the lender doesn't have to cancel PMI ahead of schedule, no matter how far down you pay the loan. If the notice you got doesn't talk about this, you should call and ask whether it's their policy to cancel PMI at 80% LTV of a recent appraisal. If they don't, and you're sure your property value has dropped, then you might want to save that appraisal fee for when you figure out how to get an appraisal for refinancing. My guess is that most banks will cancel PMI early at 80% LTV even if the property value has dropped, but I'd want to make sure.
In any case, you should send in that cancellation request. It can't hurt. The notice our bank sent talked about an appraisal, but in the end, they didn't ask for it.