If I was in your shoes, this is what I would do.
1. Tikka T3 Stainless ($699 / 6lb 3oz) (Caliber is up to you, I like a 300 WSM)
2. Bell & Carlson Medalist stock (w/aluminum bedding block) ($260 / +3oz approximately)
3. Talley Lightweight 1 piece mounts ($45 / 2oz)
4. Leupold Ultralight 3-9x33mm w/CDS dials ($500 maybe less if you can find a deal/ 9.8oz)
Re: the stock change. The factory stock from Tikka is super cheap feeling and doesn't handle recoil well. The felt recoil dropped dramatically when I changed the stock out, and it handled significantly better.
All together you will have a rifle that will shoot sub-moa / weather proof / great adjustable trigger / good glass / lightweight 7 lb 2oz w/optics / all for $1500 MSRP. Use the extra money in your budget for some top quality boots and pack, IMO these will get you farther back into the back country than a light rifle.
BTW: I did do this, I just wish they had made CDS dials when I got mine.