So, back to the thread...........? In a previous life, back in the 70's, I was tasked with trying to get DC-3's through UK C of A Airtests, eking ANY climb-rate out of them on one engine on a good day, and helping brainstorm all the vagaries of steam-driven radio-gear to boot. One a/c had severe VHF interference, which cured itself when the cockpit-to-fin ADF aerial set up such a vibration that it tore the front mount clean out of the top of the fuselage. It had been resonating in flight, obviously not repeatable on the ground. (Only mentioned because long-wire ADF aerials have been mentioned here in previous threads).
Also, any chance you can obtain/borrow another propeller for a trial fit? (Desperate measure I know, but it sounds like you're ready to try anything). Don't know about pistons, but the props on a brand-new BAS Twotter were so horrifically out of balance, such that they rocked the airframe slightly as they slowly increased speed on start-up, but ran smooth at flight RPM's. A piston-prop achieves a higher rpm much quicker on start-up, and pistons vibrate more than turbines, so you would not notice any slight out-of-balance vibration. (A set of FULLY-balanced Twotter props was charged extra!). If another prop is not available, try removing and checking the balance/blade rock on the prop you have. Might explain at least the rev-dependent aspect of your problem.
Just my pennyworth, will follow with interest.....................