Buying him out may be possible, even with a non-occupying co-signer, but you will still need cash to close besides reasonable credit and good ratios for mortgage-to-income and debt-to-income. Sometimes the reason for the divorce also damaged your credit to the point that only time will fix it.
You will also have to deal with the timing of the buy-out in relation to the divorce. Ask you divorce attorney about this. The paperwork may say he is no longer obligated to repay the existing mortgage, but the reality is that the original mortgage documents exist, and as you suspect, only a re-finance will clear the existing mortgage from his name. His homestead rights will likely be extinguished by the decree, but timing is important. You need your attorney's advice on when in addition to a loan officer's advice on how.