FI certificate to include the authorization to give course for the CPL
Semantics I know, but:
An ATO is authorised to give CPL courses. An FI, who is entitled to
instruct for the CPL, may instruct on these courses.
To instruct for the CPL an FI must have a CPL themselves (and therefore
a Class 1 Medical) and be entitled to instruct CPL students.
Part-FCL says that once an FI has 200 hours of instruction they meet
the requirements to instruct for CPL (don't even need to be "un-restricted").
To gain the entitlement the FI would need to apply to the CAA for
their rating to be endorsed for CPL instruction.
As no course, nor assessment of competence, is required I would just put
N/A in those sections of SRG1133 - a form designed (without thought) for
all possible instructional privileges.
An FI entitled to instruct for the CPL can only do so within an ATO
according to their approved course - Therefore an ATO will/should
conduct some standardisation training with the FI before they actually
instruct any CPL students.