Do you have an unused flannel backed tablecloth you could cut for batting? You could cut 2 pieces with the plastic sides together flannel sides out so you wouldn't have to worry about the tablecloth color or print showing through your coaster fabric.
I agree. I would use one of those higher end plastic table cloths (plastic on one side, white soft stuff on the other) and cut pieces from that.
Another fabric you might want to look into is duck cloth. It's supposed to block water from seeping through and it can come in different colors.