Success! Here is what I did. I downloaded the zip file here:
I unzipped the contents of that file to a folder on an external hard drive called RStudio (F:\RStudio; note that the drive letter will likely change when I plug the drive into different computers).
I already had a copy of R installed on the computer's hard drive so I just copied it to the RStudio folder on the external drive. The path to R is F:\RStudio\R-4.2.2
When RStudio launches, I changed the location of R using Tools > Global Options. The file I selected was located here: F:\RStudio\R-4.2.2\bin\x64\R.exe
After exiting and restarting RStudio, the path to the portable copy of R was found! [By the way, this is where it was crashing before. Maybe I was just selecting the wrong file. To fix, I had to delete the RStudio folder in the C:\Users\username\AppData\Roaming folder. So, this is progress! Thank you very much.]
The only remaining issue is how to keep the location for R for the Desktop version of RStudio separate from that for the portable version of RStudio. In other words, right now when I set the location of R for the portable version of RStudio it also sets the location for the Desktop version of RStudio. It would be ideal if portable RStudio didn't interact at all with the hard drive. I am not sure what the best approach would be but if this can be accomplished with a shortcut parameter we should try that first. Can you give me an example to try out?
Thanks for all your help.