You’ve got plenty of room in your case : I recently did a “portable” desktop PC build, and I just used conventional 2.5" SSDs. The reason is that those M.2 slots actually consume 2 SATA ports.
It’s cool that they now look like sticks of memory, but the packaging for things like 500gb Samsung EVOs are incredibly thin and light. The conventional packaging even protects the circuit traces inside from an inadvertent scratch with a screwdriver and finger oil, versus leaving it exposed with the M.2. drives. (not a huge deal, but damaging the device that has all your data on it is a bad thing)
I wouldn’t bother with an adapter, that’s something else to go wrong.
Here’s one. These are the same form factor as 2.5 inch SATA hard disks, so as long as your computer will take one of those, this one will drop right in.
gotpasswords has the correct answer. The SATA drives top out at 6.0Gb/s, and in practice most drives will deliver 550MB/sec. A PCIe 3.0 x4 adapter has 4x985MB/s, around 31GB/s.
Here is a review of the drive you linked, used with a PCIe adapter. According to them:
Make sure you put this in the x16 slot if you go this route.