If I'm being honest, approaches are probably the best part of my game. For me, an approach shot is anything from about 100-200, and any workable putter will do (i.e. any putter that you can throw straight no fade, anny, hyzer, powered down to flex for short distances). Rarely, I'll throw a Zone on a flex or forehand.
As everyone else has said, putter only is the way to go. Even if you love your mids, throw the putter until you can do that proficiently. Then if you still like approaching with your mids, do that. But not having that putter in your tool box is going to be a barrier for you. Practice makes perfect, and even though it might feel awkward at first, you'll get used to it. That's what practice is for--I hated throwing putters and even mids like the Comet before I forced myself to man up, figure out how to grip them, and throw them. Not liking how they feel is no excuse: there are a lot of different putters with different profiles out there.
I use a fan grip for everything, including drives. If I'm throwing the putter hard, I grip harder and tighter and do a full reach back and run up. If I'm throwing a touch shot, I throw from a stand still, grip more softly (so as to produce less spin on the disc), and barely reach back.