I am going to dive the Meg Ledges in NC next week, and I do not have an adjustable reg. I hardly ever dive over 60ft and it breathes OK there, except with exertion.
As others have said, 'NO', you do not. What you DO need, is a spool / reel and a DSMB - not because those are essential, but because certain charter operators require it for the ledge (e.g. Aquatic Safaris' website includes the following statement: '
Fossil Ledge (wreck reel and SMT required)"). The Meg Ledge / Fossil Ledge is generally a pretty benign dive (also - for me - a very BORING dive as well). But, it is somewhat easy to get lost, and surfacing several hundred yards off the boat is a PITA - for the diver, AND the boat crew.
I am a wee bit concerned by your statement that your reg 'breathes OK there, except with exertion.' Hard to tell what it breathes like with exertion. While I find the Med Ledge to be boring, there MAY be current, and therefore exertion. That is more an issue of tuning, than adjustability.
tbone really addressed the issue in his response - pay attention to it. All of my ocean regs are adjustable and have a venturi control - I have a couple of Mares regs that I use in the pool, and in quarry OW that are not, but I have taken them deeper off the NC coast without any issue. I regularly encourage divers to buy (2) adjustable second stages - why not, the price difference, in the grand scheme of things, is trivial. But, lots of people (myself included) have been to far greater depths than the Med Ledge, in more challenging conditions, on non-adjustable second stages and lived to tell about it.