When I was a flying school manager (a long time ago) this question kept coming up, and although it was technically possible, we declined to allow SFH to FAA holders for a few reasons.
1) we didn't want hassle with insurance should something go wrong.
2) there was no money in it for the school as SFH had a small margin.
3) the pilots were always unknown quantities to us and from our point of view trained to different standards
4) the was not so likely to be an ongoing "relationship" with the school.
People forget that trust needs to be built up between the school and the student, and as people do more you give you give them more latitude.