You shouldn't have to do that. Really, never install something in another installation directory for another tool.
The option to change installation folders was not available for installing when choosing "Install for current user only" option.
This will install Quarto in a User only folder, but it should still add to the USER PATH I believe.
The standalone installer for Quarto on Windows has an option to "install for all users" that requires elevated privileges
Using that will install in Program Files and make the quarto path available in PATH, for other tools to use.
The Quarto package picks up the right install folder and version number and rendering from Rstudio is good.
Glad it works, but you will encounter new issue when you will update RStudio and it could have side effect to have replace some internals of RStudio install folder. Hopefully, but we never know.