I do it all the time. Pin the backing material down with the honeycomb pins here before running the job. (And don’t move the design on the screen.)
For issues cutting through on 1/4" material - try buying a better grade of material. One thing most people don’t understand is that cheap plywoods tend to have all kinds of really nasty fillers, glues, knots and holes in it that will not cut through with any laser. No matter how many passes you use. Charred and crumbling is not the desired results.
Either real Baltic Birch plywood or the Proofgrade materials are by far the most consistent that I’ve found so far.