IMO I'm absolutely 100% positive that it should be the M777. You will be amazed at how quick and powerful this ski is. It's phenomenally stable without being overly heavy (you can kick it around if you need to get out of trouble). It has enough float to do a great job in the fresh stuff while still being burly enough to be a great crud buster if the deep stuff ain't so fresh.
The M999 on the other hand is much softer and it will excel in the deep stuff, but it's a major compromise on the hard pack that you will inevitably run into even if you're a backside skier. Also note that the 181cm M999 actually has a shorter contact length than the 176cm M777 (146.2cm for the M999 versus 155.3cm for the M777).
Note that both of these models ski MUCH shorter than advertised. Knowing what I know now I probably should have gone longer on the M999, but I think the M777 in 176cm is a "fun" length for me. I'm actually able to pop faster turns on my M777s than I could with the M666 (that I exchanged for the M999) and most of the other skis in my quiver. The ski is advertised with a 21.7m sidecut, but it skis like something with less than 18m for me. I should also note that I can run the M777s through the moguls without problem (they're not too stiff).
Can you tell I really love my M777s? The M666 wasn't what I was looking for and surprisingly the M777 actually turned out to be "that" ski when I wasn't expecting it. For my style of skiing (lots of trees, moguls, and backside stuff) I would be happy if the M777 was the only ski I owned.
P.S. I just checked the load area calculations for these skis (M777 in 176cm and M999 in 181cm) and it turns out that they're very close in contact surface area. The M777 is 1460 and the M999 is 1555. The twin tip on the M999 reduces much of its load bearing area unless you're in sufficiently deep snow.
P.S.S. I just realized that I didn't give you any specific length recommendations. For the M777 you could probably go to the 192cm based on your size, but if you're not an accomplished expert I'd step it back to the 184cm. I definitely would not recommend you going down to the 176cm on this ski. For the M999 you should absolutely ride the 192cm ski. The 181cm will be too small for you.