I wonder if a straight stitch with a teflon foot would work? It is designed to stitch over leather and the like.
Wouldn't alter the problem of needle and thread sticking though.

try the smallest, sharpest needle you can use with the thread you are using- I've found I have much better luck using a 'Sharp' instead of topstitch needles.

I just finished two t-shirt quilts this weekend and the best thing I found for those " it covers the whole block " shirts was instead of stipple / meander I chose a thread the same color as the background and slowly outlined the design in the areas that were less rubbery. It meant alot of stops and starts to get some smaller areas in the middle, but was well worth the time to avoid the frustration of trying to quilt over it !