Further digging into EASA, the AMC and specific EASA country paper, it seems that the issue of a check on each Type of aircraft for the TRE is not the case for the SFE.
An SFE would not need to do another AoC check to extend his privilege to another type.
That seems awkward as there are not many difference between a TRE and SFE?
Or am I totally out of the loop?