Never fails to amaze me how many people think PAT applies to anything electrical. P = Portable, a microwave is not really classed as a portable device.
I know we often called it PAT testing, however as far back as I can remember it was called the inspection and testing of in-service electrical equipment, all electrical installations and equipment should be inspected and tested, but it suits in many cases to split the work into two groups, the inspection and testing of in-service electrical equipment was split into two exams, as there are items with need something special to do the inspection and testing, be it a gas safe certificate, or the keys to a drink's dispenser, so the building manager is responsible for ensuring someone does the inspection and testing, and that whoever does it, has the skill.
And the inspection and testing of the installation, does not normally include any current using equipment, but the building manager can arrange exceptions. For example a hand drier in a toilet would normally be PAT tested, but the logistics of safely disconnecting it to be tested, are far easier with two people, so to arrange for that item to be done when the EICR is done, makes a lot of sense. However, the results will be recorded with the equipment register not the EICR paperwork.
Most electricians do not want to do PAT testing, it is boring, so it is common to get someone through the exam, and let them retest items, once the test parameters are entered into the machine, one only needs to enter the plant number, and the machine shows him/her what to test.
But there is no requirement to use a pat testing machine, one can use low ohm meters, insulation testers, etc, instead, specially when it does not have a plug. Needs more skill, but can be done.
However, one does not always get the expected results. Last week I had this problem, a RCD was tripping, so something was wrong, but my testing failed to find the problem, it was found by disconnecting the faulty circuit. It seems down to a faulty socket, and once I get a replacement, I will take the faulty one apart to find out what it was.
However, it has been proved that it is something with the socket, not the light which was plugged in to the socket. From what you are saying the microwave was not tested in isolation, so the tester does not really have a clue what was faulty.