Using good rings and bases, properly installed, you will be able to precisely (as in, better than 1/4 MOA) reinstall your scope each time you remove and replace it. You'll need to do proper bookkeeping to keep track of the different scope knob readings for the different rifles, but you should not have to fire any sighters at all; you'll be within 1/4 MOA of where you expect to be.
Aries' point is a good one. There's a good chance that your M14 will require one of the rings to be in a different location than your Rem 700. There are a few ways to deal with this, though we need to keep in mind your title's requirement "...without awkwardness".
You need to choose a mounting system (e.g. Redfield style, Weaver style, etc) that is compatible with the quick-release rings you choose and is has base(s) available for both your M14 and your Rem700.
The are likely fewer variations available for the base for your M14. See what's available, and see if it is possible to choose a base for your Rem700 that exactly matches the spacings on your M14 base. That would be ideal, it would allow you to simply detach the scope from one rifle and install it on another, reset your scope knobs according to your databook, then fire away.