You can build what you want for lot less than $4k.
I would start one of three ways.
Option 1. buy standard Rem PSS in 300 at around $800. Skim bed it, tune the trigger, lap the lugs and maybe rechamber if it does not shoot accurate enough (cost for that is $175-200). Total cost for the gun tweaked is $1000-1200. Scope, leup 4-16x with TMR another $800 or Swarovski/Kahles with TDS.
Option two. custom action either Stiller Predator or Lawton 7000 (IMO no logical reason for extra weight of the 1.450 for hunting gun in 300 Win) both for under $750 and good to go out of box with Rem footprint and accept rem triggers and metal. This is a hunting gun, no need for $200 trigger set at 2.5 lbs when a tuned Rem will do that for 1/3 cost. As for stocks, go to
www.brownells.com and look. HS Precision $300-500 to include the thumbholes at 2 lbs to the 3.5 lb tactical models, plus a variety of others. Barrels- Broughton, Rock, Krieger (take your pick) 1-11 twist for up to 220 grains finished at 26" with number 5-6 contour cost is max $500 chambered. lilja normally has these in stock for example. Standard rem trigger that is tuned and cost is $80. Rem bottom metal w/wyatts box $125. bases rings $100, bedding and other misc gunmithing $400.
No need to do anything else and under $2k. Same recommendation on scope. Weight will be right at 9-12 lbs depending on stock and scope.
Option 3. Go to New Ultra Light Arms, buy it built finished with a weight of 4.5-6 lbs, and normally well under 1". Absolutely top notch lightweight hunting rifles that everyone else is trying to copy. Cost is $2600 and in stock normally.
I have a Rem SPS in 300 WSM with 2.4 lb Bruce Baer thumbhole butternut stock with 26" Lilja #4 with Kahles 3-9 with TDS at is 9 lbs and will do exactly what you are asking.
any decent riflesmith can do this. does not have to be 14lb with sniper fill stock or high dollar tactical smith. Look at any of the "top recommended smiths" section on
www.6mmbr.com or one of the smiths here on LR hunting. Tthis is really not a complicated or high dollar build so do not get sucked into another $2k spent that you do not need.
BH