I concur with Beagle. A LAPL is a Non-ICAO licence therefore any training towards an initial ICAO licence is ab-initio and must be conducted by a FI(A) and does not fall within the privileges of a CRI.
In any event such traing must be conducted in accordance with the Operations and Training Manuals at the ATO. RFs do not necessarily have such manuals and can only conduct PPL training under grandfather rights. The LAPL did not exist pre EASA, so is not included in those grandfather rights.
I have a friendly ATO happy for me to work for them
These days are over, its all tightly controlled and documented now!