not an issue screwing to an EPDM roof or tpm, do like the vents, fridge, antenna etc. are fastened. Use the right angle solar panel mounting brackets, locate the spots for holes, punch through the epdm with a cutting punch to make sure you won't tear it, put a dab of the epdm caulking under the bracket, screw it down, then cover the bracket and screw covers with epdm caulking just like is done on the fridge vent, TV antenna mount, other vents.
I've done this on several trailers with no issues. I suspect this is how it is done on the trailers that come with solar panels.
I'd NEVER try to use adhesive on the epdm, tpm roof membrane to hold any panels down. IMHO it's to easy for the adhesive between the roof membrane and the plywood to come loose with air load and end up making a real mess and problem with the roof.