You can still "double-bill" in an exempt position.
Most employment contracts I've seen stipulate an expected work week of 40 hours. If I sign 3 contracts like that, now I would need to work 120 hours to fulfill the obligation. And as long as someone collects the paychecks from all 3 jobs without fulfilling the contractual requirements of those jobs, well, that is sort of the definition of theft. Money gained without performing the work necessary to earn it.