308 off-the-shelf is a great option. Fed Gold Medal Match is almost a no-brainer.
If you can find an older 700 Varmint Synthetic (either the carbon or fluted stainless barrel with an H-S stock) and add a decent trigger (CG22 or Jewell or another that might fit your preference and style) you really can't go wrong.
A 5R is a decent barrel but tends to be heavier than a Varmint (if my dinosaur memory serves that the 5R is in M24 profile). Great for shooting itty-bitty groups from a bench but a pain in the ass if you're taking it into the woods and actually moving over terrain.
You can spend thousands to get benchrester-type performance or you can shoot the thing until you wear out the first barrel, then upgrade.
Remingtons are the start point as there are so many different after-market accessories you can add as you go, or spec later. Savage 110s are supposed to be very good as well, but you're more limited in your accessories availability.
Since you've already owned 700s and a GAP you're going to have to review what you want the rifle to do -- especially since you're limited to a 200-yard range and not competing yet.
At 100 and 200 yards you can try 125, 135, and 155-grain bullets to shoot tiny bench rest groups and see what you and the system are capable of. Big-ass plates, IPSC targets, and E-types will be comparatively huge after that.
In the end it's still just a rifle (regardless of make). You want it to fit your shooting style, requirements,
and budget.
Plain old 700V, upgraded over the years (still sporting the factory barrel whish was set back a quarter-inch). Ten-shot 100-yard group, and last season's 354-yard kill.

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