That is a gorgeous lino cut!
I'm very lucky that we have presses in my lino class.
I'm wondering if spray starch on the reverse of the fabric as well as a stiff cardboard would help keep the fabric in shape, and then, a similarly stiff board (or plank) on the lino.
Sorry if you've already said. Have you laid the lino on the fabric rather than the fabric on the lino - I think you'd be able to achieve the easiest prints that way.
make my prints both ways up in the press, and we tried various methods with brayers & spoons, hands and anything else we could get our hands on when we were experimenting with different papers and printing mediums/inks/paints.