Tennessee got around the constitutional problem by only applying the residency requirement to party primaries. Anyone who meets the requirements for holding federal office laid out in the U.S. Constitution can still run for those offices, but they have to run as write-in candidates or independents.
I do not even agree with out-of-state political contributions to state politicians, no matter what office federal or state, if you are representing a state the funding for your campaign should come from within that same state.
I don’t know about the constitutionality of it. But it sounds Federalust in nature and ‘carpetbaggery’ abounds because of it.