I would do that, yes. Given that your current laptop win 7 is OEM, you'll need to reinstall with the product key that you got for the tower. Most likely, you'll need to reformat the SSD in order to get a clean install of Windows on it. Just make sure that when you reformat it, it's reformatted to GPT and not MBR, I've made that mistake many times x.x
Go into your disk management and check the partitions on the drives, when I install Win 10, it gives me a recovery partition, an EFI boot partition, and then sections the entire rest of it off. Those two partitions don't hold the Win 10 installation. Now, if you created the partitions yourself and set aside a chunk of your drives specifically for Windows, that would certainly make the job easier, especially if you don't have any other files other than Windows on it.