The Twinrocker paper is very nice, but it has a lot more texture (even in hot press) than COT-320 or Platine, so it is a bit different of a look. Great paper for small print-objects.
When coating these heavy papers, make sure that you are aggressive about drying the paper as soon as the coating solution has soaked in to where the surface is no longer wet looking. The sooner you get the solution dried, the easier the paper will be to clear. Some of these thicker papers may also benefit from the addition of PVA to the solution to thicken it a bit and retard wicking.
I have seen some nice prints on matboard as well. It must be a non-buffered board, and I can't tell you what brands or lines work well, so some trial and error with a few kinds may produce a good result with that approach. It may be possible to find a 2-ply board that would meet your requirements for weight.
Another solution is to print on your normal paper, dry mount it to another sheet of the same paper and then cut it down once mounted.