Is it possible to take checkride past AFSP training request expiration?
As far as I can tell (and from my own experience), you don't need any TSA approval for a checkride.

Looking at the regulations, 49 CFR 1552.1 defines flight training as "instruction received from a flight school" (including from individual instructors). A checkride isn't instruction, as you said, although I can't find a regulation that explicitly says that (probably because there's no real need for one).

And looking at the FAA's requirements for checkrides:


The FAA's instructions to examiners for an instrument checkride don't mention security or say that they have to check anything related to the TSA or AFSP as part of the preliminaries (see section 5-434, Establishing Eligibility and 5-446, Procedures)
There's no mention of the TSA either in 61-65F - Certification: Pilots and Flight and Ground Instructors (apart from some notes on verifying identity)


There's also nothing in the IACRA process about the TSA or security either, which makes sense because that should match the checkride eligibility instructions.

My own experience is also that it isn't relevant to the checkride. I've done two instrument checkrides after getting the TSA's approval for training and neither DE asked me for any information about the TSA or AFSP, they were only interested in my logbook (and medical etc.). I suppose there could have been some extra check behind the scenes or in IACRA that I wasn't aware of (and my TSA approvals were still valid) but I doubt that was the case.

Finally, I realize that all this still doesn't give a positive confirmation that it's allowed, so if you want to have that I suggest you call the AFSP hotline and/or your local FSDO to get a definite statement.