Generally; there are goats in all the huntable units, rifle hunters aren't much of an issue, ten weekdays is enough time IF you've done some homework and are familiar with the unit before the hunt. In other words, your best bet is to scout as many units before you apply and choose one, instead of choosing one and then scouting. Especially if you want a very large goat. But either way can work.
Goats aren't hard to hunt - they just live in very physically demanding habitat. Most hunters who don't kill either have unreasonable expectations, or more likely, aren't in good enough shape, physically or mentally, to climb thousands of feet day after day at high altitude. Occasionally, they just have bad luck with weather.
Trophy quality on goats is very different from most other critters. An inch difference in horn length can make a major difference in score. And there is always a fair chance that that will change between the time the arrow strikes and when it stops tumbling. :-( In other words - maybe don't get too picky to begin with. For many, the hair, and just the fact that it's a mountain goat is just as important. Most billies have pretty good coats, even by the first seasons, so that's not much of an issue. You have to balance coat quality with increasing possibility of getting snowed out, but there's not that much difference in their coats and it can snow ANYTIME, or not, at those elevations.
My hunt was in 2002 in a rifle area with a 10 weekday only season at the beginning of October. It started snowing on the first Wednesday, and I got snowed out Thursday and Friday. We got back to camp on Saturday afternoon and assessed the situation on Sunday. I killed a young billy on Monday afternoon that scored 43 2/8.
My wife hunted the same unit during the middle of September 2013, a couple of weeks after a successful sheep hunt that started a couple of weeks after she had a second set of stents placed in her coronary arteries. It's hard to keep a good woman down! She killed an older, larger billy than mine on her 4th day. It didn't score quite as well though at 42 even.
So, my advice; once you pick a unit, spend a lot of time in it scouting and getting in shape. Working out at home will help but isn't a great substitute for hiking and climbing above 12,000'.
One more thing, you don't have to be able to shoot at long distance. Unpredictable winds make that inadvisable despite your shooting prowess, and goats can go a LONG way in inaccessible terrain with a bad hit. It's NOT worth the risk. My shot was estimated and later confirmed at about 35 yards. My wife's, about 25 yards.
Many a nimrod heads up the trail with big plans of how it is going to go! If you jump into the deep end make sure you can swim.
Its a tough fun unit with no shortage of good billies, in the rest of the unit.
Marvelous would be a good one to hit up, also I can point you in the right direction on how I hunted it. (he probably has more info then me, I've only had one tag there) FYI it can be a nightmare if you're in there and get a lot of snow! I ended up not killing because of getting snowed out, but I passed several good ones earlier in the hunt.