I just wanted to note that my two cats came through Manchester - which I imagine is much lower volume in terms of pet transfers than Gatwick and Heathrow - with HomeAgain chips as their only chips and we got through just fine.
Also, I believe that as long as your pets are up to date on their rabies vaccines (and have had more than one vaccination previously) that the microchipping, vaccination, and blooddraw can all happen on the same day. I wish I had known that, because that would have meant that my hubby and I would have been reunited 3 weeks earlier, but c'est la vie.
As an added comfort thing, I had my vet draw up two letters on letterhead with blue pen signatures and WITH my cat's microchip numbers stating that in fact they had been chipped and vaccinated in that order. If the blooddraw for the FAVN test is drawn on that day as well, then I would have them document that too.
Encourage your vet to contact your state's USDA vet, and if they (your personal vet) aren't willing, find another vet who will. At one point my paperwork got a little farked up (like, a week before I was going to leave) and the USDA vet worked with directly with my vet to get it all cleared up.