Do TSA regulations allow CFIs to start training non-citizens before the TSA has approved their training request?
This is addressed in the AFSP Candidate FAQ, which confirms that you can start training in category 3 as soon as all information has been submitted:

How long should I expect to wait after I have submitted my training event request?
[...]
Candidates in Initial, Instrument, and Multi-Engine training (category
3) may begin training as soon as AFSP receives all required
information, including the candidate's passport, fingerprints, the
fee. You and the provider will be notified once AFSP receives the
fingerprints and you have been granted Permission to Initiate
Training.

That's also what the regulation says, but the FAQ may be a little clearer.