They’ve burned it, bulldozed it, hacked it and poisoned it. Now they want to try wasps—imported from France, no less.

The target is carrizo cane, a bamboo-like reed that’s a fearsome enemy of officers patrolling the Texas-Mexico border. Dense stands have camouflaged stash houses, half-ton steers and a caged Bengal tiger someone tried to sneak into the country.

“I’ve heard agents talk about it like it was Sherwood Forest,” said Francis Reilly, an environmental consultant and adviser to the U.S....