To my frequent frustration, RStudio treats these two control keys differently. The CapsLock key mostly works for Control, except that it unselects whatever is currently selected in the Console. So every time I want to press Ctrl-C to copy something from the Console, I have to remember to use the real control key, not the Caps Lock key.
What version of RStudio are you using? I just tried this and it seems to work as expected. (Running Ubuntu 18.04 within a Parallels VM, macOS 10.14.6 host; RStudio 1.2.1335)
I'm using RStudio 1.2.1335 on Ubuntu Disco, but I've suffered from this bug since Ubuntu Cosmic and probably back to Ubuntu 18.04. It happens with both Gnome/X11 and Gnome/Wayland. Here's a video.