This may be a strange idea, but the origin of the refusal is the FAA's stance on the SOLI'd licence. Why not communicate with them about this? Perhaps you could convince the Austrian authorities to issue some kind of statement regarding your licence to support the fact that it is a fully-standalone licence and not dependant on another country's paperwork. I know it is easier said than done, but if you can convince the FAA that your paperwork is in order, that would be the shortest route to a resolution. I'm not saying easiest... just shortest.
FAA §61.75(f) states:
(f) Limitation on licenses used as the basis for a U.S. certificate. A
person may use only one foreign pilot license as a basis for the issuance of a U.S. pilot certificate. The foreign pilot license and medical certification used as a basis for issuing a U.S. pilot certificate under this section must be written in English or accompanied by an English transcription that has been signed by an official or representative of the foreign aviation authority that issued the foreign pilot license.
The goal here would be to convince them that there is only one licence involved. You could also try to find or obtain some kind of letter or statement from the UK CAA saying that they have transferred everything to Austria and therefore there is not UK licence involved in this situation.