However, I understand that the FAA will not issue such a licence since my non-UK EASA licence clearly states that it has been issued on the basis of a UK EASA licence
When you say you "understand" what do you mean? What you are suggesting appears very complicated.
It is worth being pedantic to unravel some confusion. You cannot have an EASA licence because EASA is not a state. Your licence has been issued by a state which is a member of EASA and therefore "issued in accordance with EASA" standards. However your licence is first of all an ICAO compliant licence which is recognised world wide. It is the ICAO membership that provides the reciprocity with the USA. Before taking any action you should clarify your position with whatever country (state) that has issued your licence and/or the FAA directly. There cannot be any interest by the FAA on the issue history of your licence, but only that it is valid.