The bolt gun will be more accurate and easier to shoot.
As background, I owned a high-end AR-10 built by JP Enterprises in the 2003-2005 timeframe; I know several shooters who still use them from time to time for long-range shooting; and I have seen a bunch of people try to campaign them in 3-Gun competition (which is not really the same as long-range shooting).
The AR-10 is more finicky for accuracy. While a good bolt gun will usually shoot almost anything as accurately as the ammo is able to shoot (eg, an AI-AWP shooting Aus surplus into 1-1.25 MOA), the AR-10 will be more specific in the loads it will shoot accurately.
The AR-10 more moving parts, and those parts start to move during firing. The practical effect of this is that absolute accuracy will be reduces vs. a bolt gun and the shooter will need to be more cognizant of a consistent body position and control over the rifle during firing for consistent accuracy.
The AR-10 has a lot of reciprocating mass. There are basically three recoil impulse events during firing: the initial impulse from the bullet starting to move at ignition; the buffer and bolt carrier group changing direction when the buffer bottoms out in the receiver extension tube; and the bolt carrier group slamming forward into battery. A bolt gun has the first, but does not have the second two events. The effect of the two extra impulse events is that it can be much harder to keep an AR-10 on target throughout firing to stay on target and spot your hit or miss.
Also, historically, AR-10's have been significantly less reliable than AR-15's (not to mention bolt guns). I have heard good things about some of the "new" AR-10 pattern makes, such as the DPMS.
While in the case of an issued rifle with provided ammo, this might not matter, many long-range shooters save their brass and this is more of an inconvenience with a gas gun.
If you really want an AR-10, I would recommend GAP to build it up as you mention. They have the best chance of building one that is as accurate and reliable as you can get. However, you'll be giving up some of the inherent capabilities of the bolt gun if you transition to the AR-10's.