I called the laser shop I use for work and they said if I supply sheet $100 to cut profile. Thickness doesn't matter.
Obviously I would have to provide template or dwg and buy hardware but this give you an idea of what the laser costs.
It would be a little more to run additional ones as they would likely load a 48x96 sheet.
I think this could be made out of 3/8 Baltic Birch too and cover with speaker cloth to match drawer build, but I would reinforce on back side with 1x1 Angle and screw and (liquid nails or similar) to make extra rigid. Obviously that could be done with a jig saw.
Or you could double up on Wood around lid and wood glue and clamp.
If your handy I'm sure this can be done on the cheap and still look good.
Not trying to hurt anyones feelings that are building these, but lets be honest, a lot of guys come on here because they don't want to go to dealer.