It sure is counter-intuitive to use wood for a mechanical element that's going to see abrasion.
But wooden wagon wheels were once the norm, so there's plenty of experience of this kind of use.
Maybe the roller should be completely wear-free when contrasted with the tire. The roller gets about 10 times the contact time as the tire by size, so if it were the same material as the tire, it would wear in diameter at a rate 10 times that of the tire.
Considering this relative rate of duty, to compensate, the roller should be composed of material much more wear-resistant than the tire.
I suppose that means metal, perhaps aluminum?