HB has, I think, pretty much nailed it, with the exception that your issue is the lack of a valid CofA. The Article 33 permission is clearly where you sit, but there will doubtless be a requirement to satisfy CAA that the flight is fundamentally safe.
I think that what I would do, in your position, is set up a 3-way conversation with your friendly neighbourhood FAA FSO, and whoever mans the desk handling those at CAA Gatwick, and get an in principle agreement that FAA will issue a OpSpec D084 temporary permission, or possibly a letter saying "if it was in the USA, then we would" that the CAA would consider that as acceptable in part of the supporting evidence for issue of the article 33 exemption.
UK AIr Navigation Order is, incidentally a bit of a legal headache these days, because so much of UK aviation legislation has been handed over to EASA these days - for example with air navigation rules now being incorporated into EASA SERA. It looks to me that the UK at the end of this calendar year will leave EASA, but aim to remain in close regulatory formation with it. What will that actually look like? Goodness knows, but I'd be inclined to get your hoops in a row early, then even if you need to renegotiate in 2021, you'll be doing so on the basis of precedent.
Do be aware also that CAA are VERY busy right now with the whole Brexit thing, so getting a meeting can be an enormous struggle, but best of luck.
G