The two issues that need to be separated are:
- does the same session qualify
- can the examiner sign the license
For the session to qualify, either the examiner would need to be qualified in both jurisdictions, or the NAA being helpful, and accepting the 'foreign' form. Some more helpful than others, and some put a limitation on the rating (eg A6 only) - but it keeps a current rating on the 'front' of your EASA license.
Even if the check 'qualifies', the examiner may not be permitted to sign. The (separate) forms may need to submitted to, and endorsed by the NAA... for which there may be a fee.
Finally on the matter of who’s paying for the SIM: it shouldn't matter, especially if the content of the check is exactly the same, but it might... depending on employer.
In summary, it can work, but will vary with every circumstance of pilot, examiner, SIM approval & NAA.
Good luck, and please share the winning combinations!