Possible to temporarily move windows 10 to external hdd?

1. It would have been better had you indicated the type of PC you're working with - laptop/notebook or desktop PC. Also the make/model of the SSD and the total data contents contained on that drive.

2. Be that as it may, are you unable to wait a couple of days when you'll be installing your M.2 SSD and then simply clone the contents of the suspect current SSD to the M.2 drive?

3. In any event...ordinarily it's unnecessary to "reformat" a SSD because of the need to "install a bunch of firmware updates". And a "bunch" of firmware updates? In virtually every case the current firmware version is a cumulative version so no prior versions are required to be installed on the SSD - at least I've never run into that type of situation; perhaps your situation is unusual and you have some special reason(s) to install prior firmware updates.

4. We'll assume your current SSD that contains the Win 10 OS boots & functions without problems with the exception of the "pausing/stuttering issue it has" which apparently you're attributing to some defect with the SSD itself BUT HAS NO BEARING ON THE SSD CONTAINING A NON-DEFECTIVE OS THAT IS FREE OF ANY INTEGRITY PROBLEMS. Because if that is NOT the case, remember that when you clone "garbage", "garbage" is what you get. Capiche?

5. So if you want to go ahead and clone the contents of your SSD boot drive to a USB external drive you can do so. You will need, of course, a disk-cloning program. It sounds as if you've never worked with a disk-cloning program so if that's the case you should do some basic Google research on the topic "cloning the contents of one drive to another drive". Then if you have specific questions about the disk-cloning process you can post those questions here.

6. As a general proposition the disk-cloning process is relatively straightforward & effective. If you do go ahead using the USB external drive as the "destination" disk, you can subsequently clone its contents back to the SSD (including the M.2 SSD) when you've straightened out your system.

7. You needn't be concerned with activation of the OS as long as the current Win 10 OS has been activated.

Since this is the time you expected to reinstall the Win 10 OS perhaps all the above is moot.