I think the biggest problem you are running into is you want to sound as good through modelers and headphones at low volumes as you sound live, plus you want to do it on a budget. If you want it to sound top notch, get a Kemper. To just be able to run through your parts, ANY modeler will work. Doesn't have to sound brilliant to run parts with the band.
You may just have to lower your expectations a little maybe?