I have a customer who uses that material for the tube covers for inner-tube tobogganing at ski-hills, we've tried a lot of different things to try to get them to stick and stand up to the harsh environment these go into (water, freezing, thawing, abrasion etc.) So far the onlything we've had any success with is solvent based vinyl ink, printed, "flashed" and printed again. You can't really flash it since it shrinks, and doesn't like to stick to the platens, so we do a partial flash, then at the next station we have a hair-dryer on a stand blowing hot, this allows enough to get a second print over top and some degree of opacity.
They still lose a lot of opacity going through the dryer as the ink cures, sinks into the weave and loses thickness from solvent evaporation. We also can't fully cure them and they need to be racked for 12 hours to finish drying.
It's on my list to mess around with a few other inks like rubber inks, etc. as we are not really fully happy with the finished look, and production is painfully slow and stinks up the whole shop with vinyl ink solvent smells.
I can tell you what doesn't work:
plastisol
plastisol with nylon additive
either of the above with the material pre-treated with acetone, M.E.K. lacquer thinner, polypropylene or polyethylene primer,
regular epoxy ink for padprinting,
printed transfers
cad-cut of any ilk.
Some stick a bit, and aren't all that bad, but not good enough for this customers application. Cad-cut and printed transfers looks the best. printed transfers give you what i would call about 60-70% adhesion and durability.
If you find something that works, I'd be very interested in hearing about it.