Most of the commercially available veneer edgebanding is made of layers. I assume they do that, with some somewhat flexy glue, so that it can be rolled up. I have seen/used it as thick as 3mm, still in layers. Other than that, you can cut it yourself and get any thickness you want.
I second the statement of MFK700, huge fan of those, but it's not going to hit your $215 number though. I have never seen a router of any brand which works horizontally that will.
The MFK is brilliant though, you can control it well enough to climb cut, which will save you sanity when trimming veneer or solid wood edges. PVC is far more forgiving, you can do pretty much anything there.