Random thoughts;
No expert at all, however your problem sounds like resonance ie some part of the aircraft vibrating in sympathy with engine speed.
On the engine mount & rubbers;
- If possible it may help to get the same design/dimension rubber, in a different durometer (hardness).
- "We had a crack in the engine mount........." - this suggests severe resonance. Your engine mount should not crack, if correctly manufactured & installed.
- Have you checked the engine mount to airframe security? Seems that when you load the engine everything is under tension and no vibration. When you unload, the tension is removed and a loose/soft component is allowing movement.
- Not sure that the appearance of rubber mounts, is a fool proof method of assessing condition. If they are newish (within replacement time/hrs limits) and the correct part number, they should be good assuming bo undue stress has been placed on them.
- You have dynamically balanced the prop but have you checked prop tip tracking?
-"............... well-synched carbs both at idle and off-idle." - While low engine speed is the best for carburettor balancing, it's also important to check balance right up to Max static rpm. If a throttle cable is sticking/delivering unequal movement, it will likly show up as you increase rpm up to Max.
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