As I said before, if you are interested in consistent a LR precision rifle, go with the heaviest rifle you think you can tote. What's the difference between a 10 lb rifle and a 14 lb rifle? Answer... 4lbs. If 4 lbs is going to make or break anyone in the mountains, maybe they should reconsider hunting in the mountains or shedding weight else where. You can most certainly go with a 10 lb rifle, however, your odds of getting a good consistent precision LR rifle increases as the weight of your rifle, especially your barrel, increases. There is a reason BR shooters don't shoot whippy sporter barrels.
If you want a 10 lb rifle, by all means get one. Just don't expect the same level of reliable consistent precision performance as you would a heavier rifle. You've got to make your priorities.
On optics, there are a lot of good scopes to choose form, and the bottom line is you basically get what you pay for. I have a couple of Vortex scopes and a couple of NF scopes. I like the Vortex scopes, but I would not use them on mountain hunting rigs. I have fallen hard, full weight on my first NXS twice in the mountains and I have no doubt that a lesser scope would have been ruined. That NXS is currently mounted on a 300 RUM and performs flawlessly. You can go with a lesser scope and probably get by with it... but if you ever have a wreck like I did twice... your hunt is over at that point, or you are going to be borrowing someone else's thunder stick. IMO, they are worth the $$$$ and the extra few oz's. In fact, I think they are a good deal for the money. There are scopes with better glass, but NF glass is excellent and plenty good... good enough to hunt in last minutes of hunting light and extremely clear to see great detail at long distances. That said, in the constraints of your budget, I would go with a full custom rifle
and a lesser priced good scope and upgrade later.
This is how my 13 1/2 lb mountain rig shoots mounted with the prior mentioned 5.5-22 NXS. I was able to see the .308 caliber bullet holes @ 427 yds. It was a clear day with no mirage.