The British Embassy no longer do this. It has to be legalised in the UK by the UK Legalisation Office in Milton Keynes and then passed to the Thai Embassy in London.
To get legally married in Thailand you need an Affirmation to Marry from your Embassy to show you're not already married. I think that what you're suggesting is not possible.
All UK docs need to be legalised by the Legalisation Office in Milton Keynes and then certified by the Thai Embassy in London. It's much easier to do if you're in the UK or have someone who can help there (or pay for a lawyer/agent to do it).
She can't leave the country without a passport, so she needs to apply for a new one as soon as possible. If her extension of stay needs renewing she will have issues and she needs to avoid overstaying. As Lynnette says, it might be quicker to get an ETD, go back to the UK and apply there.
Is it the fact that she's a girlfriend not your wife, the reason? Boyfriend/girlfriend relationships are possibly not a strong enough level of commitment?