NAR / Tripoli
I would recommend getting in touch with NAR or Tripoli about your needs. They handle these sorts of things all the time. TBH if you or your range officer hasn't been certified by one of these bodies you shouldn't be launching anything larger than an E sized engine.
Both organizations offer memberships including training, certification, participation in sanctioned events, and liability insurance (some of which may carry an additional fee.) The minimal red tape stipulation will be achieved if you can join an existing sport launch which meets your needs. The biggest issue will likely be whether or not that particular launch is cleared up to your intended altitude.
NAR and Tripoli have had agreements in place with the FAA since the October Sky days and coordinate launches with the feds to make sure everything is safe and above board. If you can't join an extant sport launch, you can set up your test as a sport launch of its own. The fee you pay will be tiny in comparison to FAA fines and probably also cheaper than relocating to wherever there aren't rules.
You can, of course, choose to set everything up for yourself and apply for an FAA waiver on your own. It takes about as much patience as a moderately complicated DMV procedure. Generally speaking you can be approved for any class G or E airspace and for Class A airspace that is not in or near an air route.