What did the Felon really do? How could he (or anyone) bully Hamas into anything?

He worked around them.

The deal he cut wasn't with Hamas - it was the other countries in the region. He got them to come out in support of his deal, and to let Hamas know that they had to go. Publicly. His pitch to them (apparently) was that he would tell Israel that annexation of Gaza was off the table, force them to accept "yes" for an answer to get their hostages back, initiate a ceasefire, and that all sorts of development goodies/better relations with the U.S. were on the table. The alternative was the Full Wrath of Trump - this was now important to him, and he would punish All Who Balked Him. And He Will Not Be Balked!

So they threw their weight behind the Trump plan - which made it untenable for Hamas to hold out against it. Once Trump got Qatar and Saudi Arabia on board, they lost any realistic opportunity for a different outcome. The deal was the deal, there were no longer alternatives to be had through negotiations now that they've lost all their intermediaries....so they took the deal.

As others have pointed out, that doesn't mean they've given up their ambitions. They might yet break the deal, or they might shift their fight and tactics to a post-Trump Plan world. But they had to publicly accept the deal.