Your question has been covered pretty well so I will not add much to that. Is 1/2 moa possible for under $2K, certainly.
My reason for posting is not to really answer your original question but instead to put a light on shooting 1/2 moa at 1000 yards.
First off, its hard to shoot 1/2 moa at this range. To be honest, in most shooting coditions, 1/2 moa is not realisitic to expect. If you look at many of the 1000 yard BR match results, my main experience with this is with the 50 cal shooters association and we get match results all the time from all the matches including the world championships in Raton, NM. These are the best of the best in the 50 cal world. They shoot five shot groups for the most part.
If you could average 1/2 moa groups at these matches, you would likely win every match of the year.
To put this into perspective. The 50 cal association will award a match shooter with a screamer patch which means the group on target is less then 0.6 moa. This is considered an extremely rare and impressive feat. And thats not even 1/2 moa........
Now there are many groups fired every year that go well under 1/2 moa and generally you will see at least a couple groups a year that will break 3" ctc. But, this is extremely rare and thats from the best of the best that do this all the time with the best rifles in the world made for one purpose, to wind 1000 yard BR matches. These rifles often cost more then $6000 and its not unusual to see $10,000 invested in a rifle system.
Again, if you can average 1/2 moa at 1000 yards, you would likely win every match that year.....
Now that is not to say that all of these rifles or at least most of these rifles are capable of 1/2 moa or better easily, then we throw in the human element which in most cases will take a rifles accuracy potential up to at least that 1/2 moa range or larger.
Then you add in the environmental conditions combined with the humans ability or lack there of to read, predict and adjust for those conditions and your 1/2 moa accuracy potential of that rifle/human combo will likely increase to 1 moa or even more.
I have seen matches won with an Agg. of over 12" ctc in conditions that were very windy or with extreme variable mirage. At first glance, 1.2 moa is not impressive, but in those conditions, its more impressive.
Now, when talking about shooting recreationally, we can pick and choose when we shoot, where we shoot to some degree and we can eliminate many of the enviornmental conditions that adversly effect precision shooting. Can not do that at a match, the date is set, you go and you shoot.
So just to make my point clear, I am in no way saying that 1/2 moa accuracy at 2000 yards is not possible for $2000. That would be a very foolish comment, of course it is, just keep in mind that making a 1/2 moa group at 1000 yards is certainly dependant on the rifles ability but just as much so is your ability.
And I am also not saying your not up to it. Just realize that sometimes it just is not practical to expect this level of accuracy in the conditions your shooting in.