(1) Use the iron weights, don't go anywhere near failure, do the negative to get them to the ground. There was strong lifting before bumper plates!
(2) Consider a setup where the tires work as shock absorbers on the ground (lying on their side). Perhaps horse mats (Tractor Supply Company carries them -- among others) over the tires. You on an elevated platform of approximately the height of the tires.
(3) If you can rig up the tires on the bar to do good "bumping", then if you simply use narrower diameter plates, the tires will take the impact. For example, if the tires are bit over standard 45lb diameters, then using 25 lb plates for additional weight should give you some "slack" in the drop. You might have to "lock" the tires in the barbell more securely with this method -- though a welded iron plate to the hub should do it if you aren't beastly.
You know, you might be able to actually wrap a 45lb plate in the right sized bicycle tire. Perhaps even one tire inside another (double wrapping!).
In all cases (2) and (3), the shock on the bar is going to be pretty substantial. I would only try it with a mediocre bar.
One other thought, with the tire+hub solution, you might need a small (very solid) platform to get the right height.
Incidentally, your link was broken. Here's my attempt at it:
Straight to the Bar Bumpers from Tires Link