I wanted to give an update, in case it'll help anyone else in a similar situation.
We did end up having to give notice on our flat. If we'd stayed, we would've had to continue paying six months upfront twice a year until we finally accumulated three years of UK self-employment tax returns.
Most of the letting agencies similarly wanted six months of rent upfront continually until we had at least one UK business tax return. But we found a few agencies, as well as some private landlords, that would consider our applications based on three months of UK bank statements and our US tax returns showing self-employment income in the USA in 2016.
It took a few weeks and several applications where we got passed over in favor of more traditionally employed applicants, but we finally got picked by a landlord for a property where we were the only applicants. Hooray, and what a relief!