Other Member States have already decided, without any real rationale, that an N-reg aircraft may not be flown by sub-ICAO licence holders. To avert the risk of this becoming accepted as some formal EASA policy without any foundation, the CAA has asked the FAA to clarify the situation and the FAA has agreed to do so.
Other Member States are obliged to accept an EASA Part-FCL licence (including a LAPL) for use in an aircraft falling into the scope of Art 4(c) of the Basic Regulation (i.e. a foreign registered non-Annex-II aircraft used by an operator resident or established in the EU). That doesn't sort out the 14 CFR 61.3 issue, which I had hoped would have been covered in the BASA IPL discussions.