I agree with the purchasing concept. Buy yourself 50 or 100 new Lapua just to see if it does better. Everything I've read about tests of accuracy after annealing indicates that if it helps precision (group size) it is a very small factor. The kind of thing a benchrest competitor throwing sub ¼moa groups routinely might see a difference from, but not us mere ½moa guys. Bryan Litz could not tell any difference for long range loads using what is arguably the most precise annealing equipment made (the AMP device).
The chances are you will get much more mileage out of first making sure your gun doesn't need its lugs lapped or its muzzle recrowned or its bedding redone and being sure the mainspring is new enough not to have taken a set (less than 5 years old or so). Once you are sure the gun is in order, on the reloading front start by making sure you are reconsolidating primers properly (seating them about -0.003" beyond the level at which the anvils kiss the bottom of the primer pocket). Next, making sure your loaded cartridges have as close to zero runout as you can measure (0.001" or better TIR; lots of threads discussing this) as going from average runout (0.004" or so) to zero runout can cut as much as one moa off group size with some bullets. Finally, make sure you've tried several different amounts of bullet jump from zero to as much as an eighth of an inch or so. This seems to time the pressure peak and velocity flat spots to coordinate it with the muzzle deflection whipping phase.
After you've covered those bases and your premium brass is shooting the way you like, then go back and see if the regular stuff is still shooting wider groups. If it is, it may need necks outside turned, and if it is springy for that, then annealing may help make it workable again.
This article is a good read.