Try using skateboard truck bushings, they even come in different durometers of urathane.
I just ordered some vibration mounts from McMaster-Carr but I'll have to go around and look at the skateboard bushings.
It's been my experience that general types of rubber (such as what innertubes and tires are made of) is basically ROCK HARD, compared to actual vibration-dampening materials.
I had a mountain bike at one point (Girvin Pro-Flex) that used urethane bumbers as shock absorbers. After a number of years of no problems, the urethane bumpers turned partially to liquid while the bike was sitting inside a garden shed on a 100+F summer day. The bikes and parts were no longer being made, so I couldn't just go buy new bushings for it. Without good bushings the shocks sagged and the bike wasn't rideable at all, because the rear shock shaft (which ran all the way through the rear shock) would hit the rear tire, so I had to put something in there to use the bike at all.
I tried a few different kinds of rubber in there, all
of it felt rock-hard, with no "give" whatsoever. The best thing I found was I cut a bunch of "donuts" out of a knobby MTB tire and stacked them in there, but even that was almost rock-hard, it was nowhere near as soft as the original urethane bushings were.