Don't use rvm or rbenv on your production Linux boxes. Use packaged Ruby there.
I've used rvm in production for major sites for years with no problems whatsoever, in fact I believe it is best practise. Do you have any reasons for your preference for packaged ruby?
You'll need a compiler, anyway, once you start trying to use any number of libraries requiring compiled C extensions.
"No compilers on production!" might be true elsewhere but I can't see any reason it applies for ruby deployments.
Don't use rvm or rbenv on your production Linux boxes. Use packaged Ruby there.