WASHINGTON—Regulators would have more discretion to decide which banks to target for stricter oversight under bipartisan legislation that House lawmakers approved Tuesday, the latest front in a broad push to ease postcrisis regulations.

The bill by Rep. Blaine Luetkemeyer (R., Mo.) would scrap the current threshold for heightened supervision—institutions with $50 billion or more in assets—and instead rely on a formula the Federal Reserve already uses to determine the amount of extra capital the biggest banks need to hold....