Suggestions for a low profile that fits in my hand nicely with extra line capacity

Well I use a zillion 50th anniversary which is high speed and holds a lot of line. A cheaper option would be the tatula, tournament mg, super duty or bb1 pro. All these have very large line capacities, have high speed options and are low profile

I just started to learn this method.  So, a 7 to 1 reel is the appropriate speed?

Reel speed depends on the angler. A 7 to 1 or higher makes it easier to gather in slack quickly for a hookset or when fighting fish that are running towards you. I feel like i fish to fast with a reel faster than 6.3:1 so i only use 7:1 reels for buzzbaits, frogs, or burning spinnerbaits or lipless cranks but thats all up to the angler. I have a tendency to fish to fast so i try to use slower reels for everything. (i use a 5.3:1 for all cranks and slow rolling spinnerbaits and 6.3:1 for everything else except the techniques i mentioned above). Thays all up to the angler though and if i could slow myself down i'd probably use faster ratios

As for a reel with extra line capacity i like the lews reels and the tatula r. If im using heavy line or fishing for stripers(they pull a lot of drag and ive been spooled a few times) i use a lexa 300 but im thinking about getting a curado 300 or one of the tatula hd reels daiwa is about to release

I use a Curado E7 for my C rig rod. I also use 50lb braid so not really any need for a high capacity reel IMO. High speed reel is a must to pick up line quickly though. 

I use the Chronarch CI4HG and it's a dream to fish with. I don't see the need for a high line capacity reel. I use 14 lb fluoro with abt 1 ft of 14 lb mono on the other end and can cast it 40 yards without casting all of the line out. And I use braid as backing and my casts aren't even getting back to the backing, so this reel holds plenty of line and palms amazingly.

Any solid reel with high speed gears should do it. You determine how much you want to spend because you could get suggestions from $100 - 600 that will all so the job.

14lb. ~ 17lb fluoro  or 40lb. ~ 50lb. braid with a 12lb. leader of mono or fluoro should most likely not require a large spool ... 7:1 gear ratio preferred as you use the rod to move the C-Rig and need a high speed reel to recover line ... 6:3 to 1 gear ratio reel would also work in a pinch .