Compromise is difficult when trying to combine the concepts of Precision Shooting and hunting. Most chassis and stock designs are a culmination of experience and research into a design specific to that endeavor. Yes, there are crossovers and some are available for under $1,000. But...
Do you do most of your hunting in the St. Cloud, MN. area?
If your answer is yes, I suggest that you try to not get too excited about an all aluminum chassis. I have used them and will continue to use them but in colder weather, they conduct the cold far faster than anything in wood or composite. There are hybrids of this as well where areas of wood or composite material are placed at the hand contact points.
On the cheap end you have the Bell and Carlson stocks along with HS Precision and although there is a price difference between them, the concept is the same. There are also offerings which spring from suppliers such as Stocky's who provide similar ideas.
The extreme of this idea comes from Tom Manners with his chassis design in composite or carbon fiber stocks. These are rarely under $1,000 unless you happen on a bargain somewhere. They are not light. One of my favorites is a folder with the chassis but most will rebel against the of the stock alone.
Any one of a dozen or more manufacturers of plywood stocks will work if you find a design which suits you for 'precision' shooting and hunting. There is a range of inexpensive (Boyd's) to some of the custom manufacturers which will run way north of that $1,000 limit.
I do like the feel and handling of the GRS stocks and yes there is that void in most of the buttstocks. The Hybrid model can work and the X-Eater model has a Picatinny rail on the butt for a sled or monopod. The injection molded Berserk doesn't conduct cold as rapidly as the aluminum chassis. I did find that the mold lines had to be subdued slightly which should be unnecessary.
My suggestion would be to attend one of the major manufacturers shows such as SHOT Show or one of the NRA shows in order to get a personal feel for each stock that might suit you. Your post though seems to indicate a faster need.
Enjoy the process!