They were going to override his veto to begin with, he dealt with the hand he had to play. He stalled it and used the bully pulpit to try and get some revisions.
Such shallow surface thinkers here nowadays...
The way I see it, his stall cost him political capital and was seen as fumble as he had already agreed to the bill.
He could’ve done a pocket veto, they’d have to start all over...no veto override.