Keep the media option up your sleeve - hold off on that for now, till you've heard more from FWA and WorkSafe.
Definitely go to the GP now - the sooner the better, and tell him or her all about this that he or she is able to listen to so that you can get a referral.
Dating the letter the 5th is a stupid tactic if they're going to 'give' it to you three weeks later. It shows that they are either not very concerned about the apparently serious concerns (because written warnings are only ever supposed to be serious breaches of conduct!), or they are backdating and that also is stupid. Slipping it under your dinner plate instead of giving it to you and discussing it (not even posting it to prove they wanted to make sure it got to YOU) just so goes right against them and screams "bully" with no effort on your part, so that's just handing you more evidence.
I'm glad you've got no experience with playing hard ball, and I'm sad that you have to learn it now. The situation and your boss (not to mention manipulated co-workers) can have you feeling crazy and like you're the one in the wrong and that's what bullying is designed to do. Stick to your guns.
Mention media maybe in the next communications, that may be enough to get them to conceded a little something. Just not now till you know a bit more about where your cards are falling.
Look after yourself - get a massage, or foot rub or something. Just take some time out (I know how much these things just follow you around in your head)