I don't think that currently RStudio IDE handles this: You can't choose with remote to use with a branch when pulling with the button.
In fact, the git pull button is synced which whatever remote you git branch is tracking. If you change the remote tracking branch, the effect will change of the pull button with switch. I don't think the is a need to change all branches, for some branches why not.
However, I think you can open a FR on RStudio github repository if not already opened there to share he idea of this improved pull button
For the workflow you describe, do you use usethis 
?
There are helpers inside to help with PR and all.
See https://usethis.r-lib.org/articles/articles/pr-functions.html
I find them useful and use them myself. Maybe they may help if you don't use them currently.