Best way to get both 120V and 220V from existing multiwire branch circuit (MWBC)
You do not have a "120V MWBC". 

You have a plain MWBC.  240V-only (no neutral) loads are allowed on MWBCs. 

They even make receptacles for that. 



Note how there is no tab to break away on the visible side, since one must be neutral and the other L2.  However you can see the tab peeking out from the other side, which could both be L1 of the same circuit.. Because this is legal on MWBCs.



As far as GFCI protection:  It's an easy job for a 2-pole GFCI breaker.  Replace the current non-GFCI 2-pole breaker in the service panel  with a 2-pole GFCI breaker.  Liberate the two GFCI receptacles and reuse them somewhere useful.  

If you don't have space in your panel, then it's time to fit a subpanel somewhere. Since you're ready to do that, do that... just not out at the patio where it'll be a one-trick pony.  Next to the main service is better. After all, you have this "out of spaces" problem all over your house.  So fit it somewhere that makes sense for all your future loads.

I know adding a subpanel is a big job, "best" kicked on down the road for awhile longer... But the fact is you'll be right back here again when the wife wants an on-demand water heater or a plug-in hybrid.  In this day and age, with so many amazing electrical gadgets, a full panel is no place to be.