EPDM Roof Leak

gbopp: That method had crossed my mind. I think the depth is 3/4" and hopefully the ceiling panel is not attached, although I don't have my hopes up. There is a foam insulation layer between the plywood and the ceiling panel.

See if an oscillating multi tool might work. You can get pretty accurate with cuts.

If the interior ceiling is not attached directly to the plywood roof, use a circular saw to cut out the damaged wood.
Make sure you set it to the correct depth.

Fortunately, the roof beams are aluminum and wouldn't rot. I'm not sure what to use to cut out the damaged wood without hurting the inside ceiling. I need to cut out a square and then replace it with a square piece of 3/4" plywood.

The BEST way to handle it is to completely cut out all the bad wood. That means cutting out the plywood past where it got wet and swelled. That means repairing any structure under the plywood that got wet. If the structure wasn't wet for too long, you could dry it out and hope for the best. If it was wet for some time, you need to determine if it is still structurally sound.

Once you have examined and replaced any damaged wood, then you need to repair the rubber.

When I did this to my old trailer I bought liquid EPDM rubber than you mix in a catalyst and essentially "paint" on to the roof. If bonds to existing rubber and forms a seamless patch. That way you don't have to worry about seams.

Good luck. It can be quite the undertaking, but it is manageable. Make sure you replace ALL the damaged wood the first time though. I tried to be cheap and just replaced the plywood when I did my roof. Well the 2x2 beams that support the roof were wet and rotted (old damage before I bought it). The wet rotted wood eventually corroded through the aluminum siding, causing more issues. I had to take everything apart a second time and replace the structure that I should have replaced originally. It was much more involved the second time.