Just use adhesive then vacuum press the fabric down tight against the surface.
I would make a die that fits inside the case to retain it's shape while easily being removed.
I would make another die that presses the fabric against the rubber slightly recessing it.
Placing the assembly inside a vacuum press.
Vacuum it at a set pressure until the adhesive has cured.
Then trim the scrap away by cutting between the fabric and edge so that the fabric lays inside the plastic lip.
The vacuum should provide a uniform press filling any recesses pulling the fabric tight against the rubber.
Prepare it by removing as many gaps as you can before placing the die that presses against the fabric on top. This will prevent rips from the plastic being over extended while not allowing for wrinkles.
Best of luck with your project.