You have two routes. Does the £2,000 include the optics - assume it does.
1) hang around the shooting range. There are plenty of target / long range shooters whose real hobby / interest is getting a new project, working up a load and getting it to shoot. Identify these guys. Once their rifle is shooting well wait about 2 months and you will notice a twitch starting to appear. Through in the idea of say a 6.5 PRC as a new build, and then a month latter offer to take their old rifle onto your FAC so they can have a slot for their new bangstick. Some coin will need change hands, but the amount that does is never ever disclosed amongst gentlemen as it gets embarrassing.
2) Remington 700 SPS heavy barrel, or Howa equivalent. These can be picked for not huge sums. Basic stocks are functional but plastic. But they shoot well out of the box. The Tikka T3x would be included if we had American pricing but we don’t.
And then I would add a Sightron Target scope and spend the rest on ammo. rem and Howas can be fettled / customised as required amd as funds allow.
For 1 mile range, its not so much the rifle, but optics, ammo and in particular bullet choice and the ability to zero in and read the wind that’s critical - or so I have been led to believe.