Toward answering your questions:
I could have become allergic to it recently due to suddenly playing on it a lot?
Do you think switching to a silver mouthpiece would solve it?
No, I am afraid any alloy (not pure silver I believe) could have been in the original mouth piece so I would avoid all metal just to be sure.
Or do I need to use the wood mouthpiece to test this theory?
Yes, I would totally change to eliminate any possible metal alloy that could contribute to contact dermatitis (especially nickel and copper). AND wax the wood mouthpiece.
OR play on your original metal mouthpiece a half hour before you leave to see your physician consultant, which is OPTION 2. Changing mouthpieces is OPTION 1.