Very thin pieces can be processed by mounting them on hard wax, sometimes called black wax, and processing it, then dissolve the wax. The processing can include making them thinner to the point of complete removal with fine sandpaper or etching. The dissolving process can be very gentle, with no stress.
I’m not sure who mightbhave one in your area, but a water jet cutter may work, depending on how you secure your samples. I was about to recommend a laser cutter, untilnIbsaw the second part of your question.
Precise cutting such a thin foil can be done with pulse-periodic femtosecond laser, better second harmonic, around 500 nm, several hundreds femtoseconds pulse length with several hundreds uJ up to 1 mJ energy per pulse. The process of fast ablation (if laser parameters are ajusted correct to provide itd) does not affect substrate material at all.
You can cut using co2 laser of low power and fast travel using a galvanometer laser engraver for sample, do a lot of low power passes to avoid damages, you can use inert gas like Argon to help avoid damage.
Or
Uses a pressurised water cutter with some kind of narrow nozzle.