...and I won't fly unless I feel well enough."
Son, you're going to have to learn a few things about aviation. One of them is that you go fly when they tell you to fly. There's none of this, "Oooooh, I have a headache, I can't fly today," crap. When it's time to go, you go. Pilots who are sickly and always trying to weasel out of flying generally don't stay employed very long.
Years ago I was having breakfast with a friend of mine. At the time he was a young guy with a fresh Commercial certificate who wanted to make helicopter flying his career. I was telling him about flying out in the Gulf of Mexico for PHI, which I did for thirteen happy years. Nine of those years involved living on an offshore oil platform for my entire seven-day "hitch."
"What happened if you got sick during the week?" he asked. A fair question, I guess.
The answer is: You don't get sick. If you wake up feeling bad, you put your big boy pants on and you go fly. See, PHI didn't have a bunch of pilots sitting around at each base just waiting for somebody to fall out sick. Virtually all pilots had job assignments. Those unassigned (i.e. "pool") pilots were generally the newbies who didn't have enough experience to fly for certain oil companies, like Shell for whom I was assigned. So a pool guy couldn't have come out and taken my place even if he was available. Pool pilots generally flew the pop-up charters that came in every day.
Not only that, there usually weren't any unassigned ships at the bases to ferry a relief pilot out.
And what would I do for the day, stay in my bunk?
No, you get up and fly. Luckily the "field ship" that I was flying never strayed far from the nine platforms that I serviced. Because there were days that I'd do a flight, land back at the main quarters platform and immediately hit my bunk until the next flight.
(That young pilot I spoke of is now an experienced Utility pilot doing powerline work, fire-fighting and lots and lots of "long-line" stuff. He's amazing and I admire him greatly for his achievements and skill. When he's out on a job at a remote site, he gets up and goes to work every morning. He knows the score. He does not get sick.)
Even at my current company, doing this cherry-drying thing, we have just enough pilots to cover the contracted ships. If one pilot falls out sick then that ship doesn't fly and the company doesn't get paid. Have I ever hovered over cherry trees when I would much rather be back in my bunk because I was sick? Not that I'd admit to the FAA. But between you and me, yeah, of course, all pilots probably have. There are plenty of areas of aviation where pilots work in remote places where no replacements are available, and calling in sick would put a MAJOR crimp in the operation.
A company will take this into consideration when deciding whether to hire you. If there's even a possibility that you might not be able to perform your duties due to an illness, you won't get hired, simple as that. They'll hire the healthy guy. Sorry, that's just the way it is. Not saying you *can't* do it - God knows that anything is possible.
But as I've said, you're asking the wrong people. You need to ask the FAA if you'd qualify for a Class I or II. In my *opinion*, the fact that they gave you a Special Issuance Class III does *not* mean that a Class II is guaranteed.
Now, if you're asking me (as the guy who hires and fires pilots at my company) if I'd hire you given everything you've told us so far, my answer would be no. Why would I? It's just too risky.