PIR isn't going to work through glass, I don't think. A microwave sensor should. Or something machine-vision based (in as much as machine-vision things ever work).
AIUI you have to get up close and personal to read the RFID chip inside a cat, so that's a non-starter without a cat flap type opening to constrain it.
A reflective optical sensor might work, subject to the albedo of the cat and there not being too much sunlight. I've been mucking about with a cheap reflective beam sensor (of the type designed for controlling motorised gates and similar) for detecting BHPCers with dodgy tags[1] riding through the finish line, and while it's intended to detect the beam being broken when it's bounced off a retroreflector up to about 10 metres away, it also does a reasonable job of detecting the reflection from random objects[2] in the under-half-a-metre range. You can get small industrial sensors designed specifically for this sort of short-range sensing: think hand-drier.
Obviously a beam-break could also work, but would require Stuff be mounted outside. See also pressure sensors, capacitive touch and so on.
[1] If you could attach an Impinj UHF tag to the cat and mount a circular-polarised antenna above the doorway that would work really well, albeit a couple of orders of magnitude over budget.
[2] Chairman Al's legs, chequered flags, that sort of thing...