That's an interesting one. If your privacy settings don't prevent this then you're fair game for being viewed, albeit that Facebook settings change so often and are so confusing it's hard to keep up.
Most people do tend to assume that if someone pops up in "People you may know" this means their profile has been viewed but it's not necessarily the case. Facebook makes some terrifying associations through processes it largely doesn't reveal. I have heard of a case where it connected two people via "People you may know" where one had an injunction to keep the other away. The person against whom there was an injunction needed to get to court to prove it was Facebook algorithms and not them looking at the other persons profile.
The reason I am commenting is that people do need to be aware that this is not just confined to GPs. If it's in the public domain then it's fair game.
Whilst I am easily found on Twitter I am pleased to say there's no trace of me on Facebook and nor will there ever be. I understand why people use it but such things always come with risks attached. Nothing to stop any healthcare professional from looking you up.
However, there is at least a counter argument with appeal hearings if one member has looked you up and the others were not aware of this. Unless the information discovered is revealed at the outset and put into evidence then the tribunal will always err in law as they must always start with a level playing field. One member of the panel cannot have more knowledge of your case than others.
A good example of a potentially legitimate use might be where you tell PIP you can't walk more than 20m to your nearest shop and a tribunal member looks it up pre-hearing on Google Maps and discovers it's 50m. Provided they raise it at the outset and explain how they came upon it so that every person in the room is aware of it and has an opportunity to address it then it's probably legit. Ditto if a healthcare professional did it or indeed a DWP decision maker.
Back on topic - yes, second opinion definitely the way to go. Not necessarily from a Fibromyalgia expert though. Often useful to simply ask for a referral to different types of expert. You'd be surprised how often it opens up a completely different line of enquiry in terms of diagnosis.