Lapua or not, once the case is fired and resized it gets work hardened. The paper clip analogy above is a good one. It'll only take so much before it gives. Depending on your setup, it could go several firings or just a couple.
I anneal after the first and every firing. I don't know when it'll work harden enough to effect neck tension. But I know if I anneal the same way each time, it'll stay consistent.
Id imagine annealing effects spring back as well. Which will further effect tension.
I run a benchsource and can't see a reason to upgrade to an AMP. I don't shoot huge volume and have enough brass to keep me shooting. When I get half way through a batch, I will run it through my process. Anneal, size, trim chamfer and debur.