I simply wonder whether Hantek can be so bold as to write 300V next to connections that can only handle 30V.
I doubt it very much.
Input impedance is probably 1M Ohm so 300V isn't likely to damage it - it's less than 1mA of current.
CAT ratings include transients though. That's where the doubts are.
This is academic though. Just
don't use 1x probes for this check the switch and check a lesser voltage before probing to make sure the switch isn't broken.
Also: Get some fixed x10 probes (or even x100 probes) if you regularly want to look at mains AC. Reduce that 300V to 3V.
Even better: Get a differential probe and avoid the dangers of earthing as well.