Keep:
After rereading your answers and checking here:
https://www.caa.co.uk/Commercial-ind...ered-Aircraft/
I see where the "aircraft build statement" is coming from. But to imply getting this statement from the OEM without a physical inspection sounds to me like a NEW aircraft requirement not a USED aircraft requirement. And this "build statement" appears to be an administrative item within the CAA as I don't see anything similar on the EASA site for an Export CA.
Further down on that web page it mentions scheduling a CAA survey of the aircraft. Seems to me that survey would be sufficient to issue your C of A and provide the required conformity statement (TCDS) as that is how it is done here on a used aircraft.
Another option that may still be available (not sure) is to bring a FAA DAR over and have him complete the Export (aircraft wise only) for you since the FAA is the "importing authority."
Regardless of how you obtain your C of A, someone needs to physically verify the aircraft does meet the FAA TCDS before it leaves. If you are the seller your C of A is only half the process in getting the aircraft available to the buyer in the States especially if using an escrow service for the purchase.
When the aircraft hits the US the buyer will need a DAR to issue a FAA Airworthiness Certificate which requires the DAR to perform a physical inspection of the aircraft and records prior to issue. If he finds something amiss now, you could be screwed as the new certification process takes place only AFTER the aircraft has a N reg issued and the G reg is de-registered.
Then the real fun begins as most escrow services will not release the purchase funds until the aircraft is certified in the destination country.
Good luck... again!
W1