The solution may not be exactly what you are hoping for, but it would be to not use cloud hosting designed for traditional documents, but to use git and GitHub or something equivalent.
I'm not 100% sure which auto-save options you're referring to, but the ones you can control you'll find in Tools >> Project Options (or Global Options, if you'd like the setting to be global)
It's currently not possible to disable RStudio's behind-the-scenes auto-save mechanism. Would you mind filing this as a feature request at https://github.com/rstudio/rstudio/issues?