Does anyone know if Capture One supports LUTs?

I usually apply them when inside of photoshop, but I do like the way that C1 handles color better and can't help but think that it would be a PERFECT addition to C1.

So the short answer is no.
Long answer is with a 135£ piece of software for the LUT to ICC conversion.

I'll just stick with doing the treatment in PS after I make my C1 corrections. Would've been great to handle all aspects of the coloring process inside of C1, but alas...

If you really want to stick with C1, you can probably try to convert LUTs to ICC files using OpenColorIO (which is free):

http://opencolorio.org/userguide/baking_luts.html