Based on some more reading I was doing yesterday, am I correct in assuming that we could apply for a visa to "activate a green card" (which since we have been 'married for > 2 years' would be an IR1), go to the US is successful, activate the green card, and just continue to return at least once a year to keep the card active for up to 10 years?
We won't be pursuing this without professional advice, because on the second refusal, we have discovered ICE reported it to the FBI as an arrest for trying to enter the country without proper paperwork. It is a long story of how we thought he was ok to come in, but that confusion has obviously complicated the matter. Coupled with the fact that nobody actually knows how applications like this will be processed yet, we have some big challenges in front of us.