If you want the external USB hard drive/SSD to contain both the bootable macOS USB installer and the full macOS install for a fully functioning external macOS boot drive, then you need to properly prepare the USB drive to support both. Erase the USB drive as GUID partition and MacOS Extended (Journaled). Then you will need to add a new (second) partition. Make sure the second partition is just large enough to hold the macOS USB installer. Make sure that the first very large partition is erased as APFS (top option) since this will be for the full macOS installation while the second smaller partition is for the USB installer which should be erased as MacOS Extended (Journaled).
Follow Apple's instructions for creating a bootable macOS USB installer making sure to use the volume name for the second partition (the smaller of the two partitions) as the destination when using the Terminal command line for creating the USB installer. When finished, Option Boot the USB drive as before. Now when you click on the "Reinstall macOS" option from the installers menu you should now be able to select the first larger partition as the destination for the full installation of macOS.
FYI, you will now be stuck with an extra smaller partition. It may be possible to delete the smaller partition used for the USB installer and merge it back into the Container for the full macOS boot volume, but macOS can be finicky when modifying partitions. Modifying partitions like this does risk damaging the other partitions on the drive which in this case could possibly damage the new macOS full boot drive if something goes wrong.
Why not just purchase a USB stick to use for the bootable macOS USB installer as that will be so much easier especially if the external drive is a hard drive as opposed to an SSD since you will end up thrashing the heads of the hard drive back and forth during the install process which will make the process take a lot longer than if you had used a USB stick for the macOS installer.