When Palmer Luckey’s Erebor Bank launched earlier this year, it made an attractive pitch to lure new crypto customers: It would convert their stablecoins to cash for free.

But Erebor wound up pulling the offer after several months, after sophisticated crypto trading firms sniffed out a way to make an easy profit out of the offer—at a cost to Erebor. The episode demonstrates that the young bank is still figuring out how to manage the risks of serving the cryptocurrency industry, a world of fast-moving financial flows, colorful characters and unusual business models.

Block, the fintech firm cofounded by Jack Dorsey, said it has applied to federal regulators to establish an uninsured national trust bank called Builders Bank & Trust.

The company has sought approval from the Office of the Comptroller of the Currency to set up the proposed institution, ...