Your candidates for solutions to your problem are mass (as Nik has specified), or shock absorption *as you have already mentioned). Or a combination of the two.
Available shock absorbers (e.g. screen door closers, SUV rear window and car trunk lid supporters, motorcycle shocks) would have to be somehow adapted to your use. That difficulty, the trial-and-error approach necessary with the risks inherent thereto, plus expense might rule this out.
Next would be mass. If you check your local building codes, you can find the allowable floor load, both dead and live, in pounds per square foot in your area. (Just as an image of what that might consist of, think of several four-drawer file cabinets [each of which when loaded can be very heavy], arranged in a circle like petals on a flower. It is not at all unlikely that these file cabinets could violate the allowable residential floor loadings, especially for a load in the middle of the floor.)
So, I would think that the best and lowest cost solution for you would be to get six 94 lb. sacks of Portland cement, build a wooden case or box for each (3/4 inch plywood bottom and top, 2x6 sides, screwed together, with flat pieces of plywood taking up all of the space between the top of the sack and the bottom of the lid, line the boxes with thick visqueen to keep cement contamination into the room to a minimum), and place one of these under each of your six legs. Look out for your rough boxes poking holes in the visqueen, snagging rugs, and tripping people moving around the trampoline.
That would be 235 lb. working against around 600 lb. You might stand a chance. (You could try adding six more boxes stacked on the first six for more mass.)
Going back to shock absorption, maybe building the boxes like above and placing one or more layers of foam material in each box (arriving at the number of layers by trial and error), with a plywood "piston" on top of the layers, and each leg pushing down on the piston, maybe you could get enough damping that way. Maybe the piston would be best is doubled (two pieces of plywood to total 1 1/2 inches thick glued and screwed), and a candidate for the foam might be those cheap, interlocking foam panels at Harbor Freight or one of those big lots stores.
Good luck solving your problem.