I'm not sure a laser cutter would be the best option for your requirements. Personally i'd be looking at a router, possibly with an oscillating knife setup for cutting closed cell foam inserts. Have a look at this video I found, is this the type of thing you want to do? : AXYZ CNC Router Knife Cuting Foam - YouTube

I can't say i've ever seen a DIY machine with an oscillating knife, but you soon learn on here that most things have been done before.

Si.