You can actually request a rectal thermometer.
You can also request that they not charge you, or write you a script for morphine, I suspect you will be about as successful in any of those requests. The nurse who takes your vitals is not there to tickle your perv bone so to speak, and they are not going to waste time finding a rectal thermometer, and if necessary bringing in a chaperone so you can drop your drawers and waste another 3 minutes waiting for the thing to register. Just not going to happen.
The last time I was at a doctors I don't think the nurse spent more than 3 minutes introducing herself and taking all of my vitals. They have a lot of duties and helping give you a wet dream that night is not one of them.
In this day and age, if you want a thermometer shoved up your ass, and I am not saying that is a bad thing, post an ad and find a playmate.