I'm not an expert on rubber roofs but I do a lot of diy projects. Without pictures I just have to make some guesses.
RV Rubber Roof I found this website about a person that recoated his rubber roof himself--it goes step by step. He seemed to have the same problem so maybe it can help you. He has a three year follow up that says all is ok---so it may help.
As far as the dampness under the roof you must find the least intrusive way (if you intend too preserve your roof) to find out the extent of dampness and has it left any rot. Rot equals weakness, dampness does not necessarily mean weakness but it must be dried or it may turn to rot.
Can the edge of the rubber roof be lifted enough to let you see or feel the extent of dampness? How far back into the roof structure does it go. If so then I would work from the edges so I could refit the seam when I am done. Lift the edges and see what you have going on keep searching until you can find where the wood is dry not damp.
Can any of the roof be exposed from the inside---removing a vent or under side of a cabinet---anything to gain access.
Then you must dry the damp wood with fans or heat perhaps a tarp over the effected area and a fan blowing inside the roof under the rubber if some heat can be appplied so much the better.
If you have rot then the job just became bigger---depending on how much rot. If you can access the extent of the rot and carve it out and replace it with new wood glued and retain roof integrity then you dodged a bullet---if not then you have to have access.
It may mean cutting out a square of the roof---and I just do not know exactly what that means. Is the roof glued to the roof frame? If so then it would be tough to get off. If it is not glued it seems maybe a patch the smallest possible could be cut out allowing access then the cut out patch which can be put back down using eternabond and then coating the roof.
Since I am not an expert I am going to stop here and see if anyone comes along to concur or give better information. With the info I have I just provided a description of what I would be thinking if I were you.