Over the last couple of years I've tried different things on canvas and leather strops. In judging by face feel for me, I felt that not putting anything on the strops gave the best results. I like a buttery smooth blade. I even quit using water stones and I'm now exclusively with oil.
To answer your question, I don't think the crox will bleed. Of course that depends on your webbing material. Before I got my current one I went to the army navy store and got a 3 inch wide canvas web strap. I was only about $5 I think. Probably went up a little since that was about four or so years ago.
Cheers,
John