Generally beyond the scope of PAT but could be done during fixed wiring checks and yes they should be tested.
The inspection and testing of fixed/hard wired appliances may be beyond the scope of PAT, but is most definitely within the scope of in-service inspection and testing, as evidenced by the Code of Practice for a long while now.
So it doesn't necessarily mean it can only be done during the fixed wiring I&T, nor does it mean that it can't be done during in-service I&T.
The important thing is that whoever is carrying out the work is competent to do so.
So in-service I&T operatives/companies shouldn't dismiss fixed appliance testing out of hand ("it's not portable!"), unless they are not competent to do it.
Nor should fixed wiring testers refuse to do it because "it's an appliance, innit?".
It's down to the duty holder of the premises to contract the correct people to cover all the work needed, and agree what their remit is.
Unfortunately, a lot of testing operatives think that because they have their 2377, and it's covered in the CoP, they must
test these sorts of appliances.
A step back, and a brief self-assessment ("Am I competent to carry out this work?") is definitely in order in some cases.