My suggestion is to buy some liquid EPDM, I've used a product called "Liquid Roof". It is a 2 part system that you mix together and apply like a thick paint. When it drys it forms into EPDM rubber, just like the roof membrane on your camper. It will adhere to your roof as well.
I would cut back the rubber away from the soft wood. I would then cut out the soft roof sheathing, making sure you cut back until you have good wood. Check the 2x2's underneath... hopefully they are in decent shape. If they are rotted as well, you gotta replace those too. Once all the bad wood is out, let it dry out completely. Then replace the wood with similar quality and dimension and replace any wet insulation.
Once the roof sheathing is back on, make sure the adjoining rubber roof is clean and dry and mix up the Liquid Roof EPDM. Paint the liquid EDDM on the exposed wood, making sure to overlap the existing rubber roof. Put on sufficient layers to build up the material to the correct thickness. Let it dry for a couple days.
You will once again have a seamless rubber roof and will have repaired the soft sheathing.
While you are fixing that, thoroughly inspect the rest of the roof. If the caulk failed in one spot, it could have easily failed in another spot as well. Make sure you fix EVERYTHING the first time. After the roof is fixed and solid again, I would recommend putting Eternabond tape over all four edges of the roof and overlap the siding. Caulk requires constant inspection and re-application. Eternabond tape will last longer than your trailer. Do it once, do it right, enjoy camping!