GFPT approval to fly
Is not so strange to have a GFPT fellow who has done their training somewhere else.. what is strange, I agree, is that the student wants the instructor to make the weather assessment.
The Day VFR syllabus requires that the student pilot can interpret weather for the local area, and is familiar with local procedures.
You need to satisfy yourself that the student can meet these obligations with respect to the local area. It doesnt matter what the other school has in their ops manual - what matters is whats in yours (and in the regs, which should be reflected in your ops manual).
(Of course as the instructor signing them out, you will always have the eye on the weather and have final rights to allow / deny the flight.. that is not in question).