I am thinking to buy new disks for my DS413 and run SHR-2 until either I find my NAS too slow or if it fails for whatever reason.
Is it possible to just buy a new Synology NAS and exchange the disks?
You'll want to use an identical hardware raid controller. Getting a new but otherwise identical NAS and transferring the disks in should work. Getting a newer NAS from the same company might work if you are lucky.
A more expensive but fool proof option is to buy a new NAS complete with disks, and transfer the data to it via the network.