As part of the order, Trump cannot do business with NY banks...
“As part of the order, Trump cannot do business with NY banks...”
That part of the judgment has been stayed pending the appeal, even without the posting of a bond.
What isn’t stayed is the right of the judgment creditor (the State of New York) to start enforcing the judgment. That means seizing Trump’s assets to satisfy the amount that he currently owes. (He owes it unless and until it’s reduced or eliminated on appeal.)
So Trump is free to deal with banks and surety companies. His problem is that institutions writing appellate bonds want collateral. In particular, they want collateral that they can easily liquidate if need be. That means cash, stocks, and bonds. They aren’t keen on accepting real estate, and that’s what Trump has a lot of.