Thanks so far!
No, I am not using the lite SAS, not sure whether I even noticed that in the store! I'll give that a try. Trimming the SAS so that there is only the outer shape sounds like a good idea, then the inside of the applique would be completely without web. I'll try that, too.
I've seen the suction cups, but somehow I think even using one of those or tape etc. wouldn't achieve what I am looking for. My problem is not so much that the ruler slides off the fabric because it's slippery, but it moves when I cut even though I press on it hard. Maybe I press the cutter too much against the ruler?