Hi Josh..
Since you went to Gatwick, I take it that was for initial Class 1 medical.. Hopefully this disappointment has occurred before you've spent/wasted/invested a lot of money on training towards a professional licence.
Anyway, here are my thoughts ( please note that I spent many years on the aircrew selection and recruitment side and encountered many Class 1 issues similar to your own.)
1 Yes, there are grounds for appeal but you may well not be successful and it won't be resolved overnight.
2 Let's be brutal here... just take it on the chin..OK? Right, at 5'7" you're clearly built like the original brick s***house.. you're a very big guy and in a competitive interview/recruitment situation you are going to be remembered.. but for all the wrong reasons.
I'm sorry Josh, you'll be much better off by working to meet the CAA's BMI criteria than appealing their decision... and your career prospects will improve too.
Good luck and best wishes,