Even with the best balance, you have a large single piston trying to maintain RPM on what, 250# of spinning flywheels... Whatever you mount it to has to be flex free, otherwise the torque pulses will move it every time the engine fires... If it moves worse with load applied, something is giving in the structure.
Quisp
When you got your engine, was it complete, or did you put on the wheels and head? Do your flywheels have cast in counter weights? I know george and Joel got in a batch of center balanced wheels at some time a few years ago, that didn't have counterweights. I know, I got a set with my kit engine. This wasn't a big deal as I was going to balance it anyway and it gave me a clean slate to start with. I ran it a bit before I balanced it, with the engine bolted to generator frame with the generator on the frame also(about 1000# total weight). Even without counter weight, it didn't move all that much setting unsecured on my garage floor.
You description sounds like flex in the game somewhere which is storing and releasing energy. I like the cast concrete block poured up thru the floor idea personally, but whatever method really needs a rigid steel frame that engine and generator bolt to. I had some pretty heavy wall steel box available to build mine, but even it has a little flex. If I was building another I would use heavy "I" beams with heavy wall round pipe welded in as cross members. If you had a rigid frame to bolt engine and generator to, this will take care of the majority of the torque forces. Even if bolted to the floor, this would also spread the vibration out across a larger portion of the structure. I think even a rigid engine frame would still have problems in your situation though, because if it can flex, it will flex, and your floor does flex...
As far as movement and vibration go, mass is your friend. A body in motion tends to stay in motion, while a body at rest tends to stay at rest. The greater the mass , the more force it takes to get it moving. Your vibration and torque forces are limited, so the greater the mass of the whole assembly, the less it will move with these forces applied.