Are you only asking if it’s laser safe, or are you interested in ideas for other options?
Are you needing to cut complex shapes? If you’re considering other approaches, knowing the shapes and sizes you’re trying to get would help. For instance, a guillotine would be good for some kinds of shapes and a hobby knife (X-Acto) would be good for others.
Does it stiffen up when it’s cold? I know some places just have liquid nitrogen on-hand, if it’s easy enough for you to get your hands on some, maybe freezing it would help.
Similar to sticking it down to something tacky, maybe you could stick it down with something mechanical. I’m thinking a sheet of something with a ton of tiny little spikes on it, something like a sheet of the hook half of Velcro. Unfortunately, I don’t know if sheets of tiny little spikes actually exist.
If you are left with cutting the shapes by hand, maybe you could go the sticky mat approach, but make sure the glue doesn’t touch the final shape. Maybe you could place the Power Mesh on top of a sheet of paper, then put another piece of paper on top of that that has been cut into the shape you want; this piece will act as a mask. Then use some kind of spray adhesive and spray the whole thing down. Before the glue dries, peel off the mask. Allow the rest to cure, then cut both the paper and the mesh with scissors (or whatever). Of course, you’d want to offset the mask by some amount to give yourself a buffer when cutting (and a buffer to keep the glue from seeping into the part you’re keeping). With this method you could prolly use any ol’ glue, I bet even spray paint would work.
Roller knife sounds like a good idea to me, but will a vacuum table hold a mesh? I’d guess it wouldn’t.