Related to Start Rstudio Server session in conda environment
and How to manually set all required environment variables in RStudio Server Open Source
RSTUDIO_WHICH_R is not an environment variable that R itself uses. It is used by RStudio in some case. R and RStudio are not the same
You have to keep that in mind when trying to customize your way. You need to try setting this env var on your system, not on R side.
Again, the behavior you observed is due to how a conda env behave, it is no issue per-se with RStudio IDE.
When you activate the environment conda change things on your system for it to work and find the correct R version.