FB- sounds like you have it figured out. Just be sure to wipe down the end of the thermometer with a cotton ball/alcohol before storing to clean it.
I wrote on my pet thermometer “PET” with a Sharpie marker. With a bunny you don’t want to insert very far- and some thermometers are actually flexable (though plastic) and that can be helpful- as if the bunny starts to struggle- I would recommend removing the thermometer as you would not want to cause any accidental damage (possible not likely).
I would say for taking a bunny temp- you use petroleum jelly and insert the thermometer past the metal tip. I like digital as then you know when it is done and the results are clear.
Good job for doing it successfully- I know it can be very daunting. I am so used to doing gross stuff- that I have had hubby, friends, and barn friends either look at me weird or am glad I am there to do the “dirty work”.