I'm not an expert in RStudio Server, but it's my understanding that it runs as its own user (by default rstudio-server). So that probably explains why it has a different PATH variable to you, sitting in an interactive bash session. You can also override the R used by RStudio Server, although I don't think that's what you're looking for.
This conversation on setting the environment variables might help, though!
I think you really hit on something. I creating a .bash_profile for the rstudio-server user to set the path. That didn't work. I also tried using the server-user=myuser method for my user name as done here. Neither loaded the environment. So, I'm still working on it. But I think you correctly identified the problem.