I would get the bolt gun.
As a practical matter, I think you would be best off getting a bolt gun. Here is why.
1. Cost. You can buy a nice tactical bolt gun for about $600-$800. I bought a
Remington 700 VS (varmint synthetic) which is the exact same gun as the Remington 700 PSS (police) with a non-tactical stock. The AR-10 is going to range from $1400-$2100. If you chose the bolt gun, this will leave you the perfect amount of money in price difference to get a quality scope that will enable you to reach out and touch targets at 300+ yards. I bought a Leupold 6.5-20x50mm Long Range Target scope for my 700.
2. Magazines. From the
Armalite website; 5 round magazine $35.00, 10 round magazine $30.00, and 20 round magazines
$85.00!. Ask yourself if that extra 15 rounds is worth all that money? It might be.
3. Theory. In most cases a good bolt gun is going to shoot better than a good semi-auto. You can get semi-autos that shoot more accurately than bolt guns, but it costs lots of money. I mean lots. If all you want to do is shoot the best groups as possible at 300 yards, I would go with the bolt gun. My Remington 700 shoots great. I consistantly get sub-moa accuracy out of it with the only modification to the actual gun coming from adjusting the trigger myself.
Whatever you decide, I think you would be happy either way. I just think you are going to get more satisfaction out of a bolt gun with less money spent. The key is to buy the bolt gun and then save up and eventually buy the semi-auto too. And in that case, consider
Springfield Armory's M1A. That is the
rifle combination I have and I am really happy with it.