Updating the OSX on my OSX Base System disk (Mac Air)

Mavericks and any macOS version of 10.12 or earlier will not be able to work with an NVMe SSD.

If your laptop has never had macOS 10.13+ installed, then you cannot use a third party NVMe SSD since the laptop's firmware doesn't know how to communicate with an NVMe SSD. You will need to install the proper Apple SSD for this specific laptop and install macOS 10.13+ to the Apple SSD before you can attempt to physically install an NVMe SSD. Maybe you can use a standard M.2 SATA based SSD with a Sintech adapter (not sure if it works with a SATA based SSD), but there are no guarantees this option will allow you to install macOS 10.13+.

If your laptop has had macOS 10.13+ installed, then you will need to boot to a macOS 10.13+ installer either through Internet Recovery Mode (Command + Option + R) or from a bootable MacOS 10.13+ USB installer. If the laptop had macOS 10.13+ installed at some point in the past and you are are booting to a macOS 10.13+ installer, then perhaps the SSD adapter you are using is incompatible if you are using a standard M.2 SSD (the Sintech adapter seems to be the recommended adapter for using a standard M.2 SSD).

Unfortunately the system firmware updater included with macOS 10.13+ installer requires a properly formatted internal drive which means it must at least be a SATA based PCIe SSD (not sure is a third party SATA based PCIe SSD will work). You need to make sure that the Apple SSD is made for that specific model laptop. While Apple SSDs for some other models may fit the SSD slot macOS can detect the SSD is not made for that particular system and may not work properly or even allow the SSD to be used even though a non-Apple OS may work completely fine with that same SSD. Apple expects things to be the way they shipped from the factory and any deviation may cause unexpected side effects. This is from first hand personal experience.

Perhaps there is a way to separate the system firmware updater from the macOS installer and make it work from an external drive, but I don't have that information and have never seen any instructions to do so. Unfortunately you are stuck trying to either use an original Apple SSD or trying a third party SATA based PCIe SSD (OWC made them at one time for certain models not sure about a 2013+ laptop, or try an M.2 SATA SSD with an appropriate adapter -- Sintech is the recommended one, but I don't know if it works with a SATA based SSD).