If we have an individual that is a resident alien but has no U.S. TIN -- it sounds like we need to backup withhold or just not pay interest.

If we do backup withhold, how do we report that when we have no number to tie back to the individual?

A resident alien without a US TIN is an oxymoron. You needed a W-9 completed at the time of account opening as they are a US Person. A green card resident alien is required to have a Social Security Number. Talk to your legal counsel for your next steps.