macOS 15.4 update failed on external disk

I've got a maybe somewhat odd setup here. MacBook Pro M2 Max with macOS Sequoia 15.4 running on the internal SSD (APFS, encrypted). In addition I have an external NVMe SSD (Samsung T7) where I've also installed macOS Sequoia to be used as StartUp-Disk if need be. With that setup I keep my private and my corporate macOS installations physically separated. "Just" using the same MacBook Pro to run them.

Now I tried to apply the latest update from macOS 15.3 to 15.4 on my external SSD. The update shows, the external disk is set as StartUp Disk and the update process starts. It looks like it does the update with multiple reboots and all. But after it's finished and log in to the macOS it still shows the update 15.4 as available. There were no apparent errors during the update process. But I noticed that wen I log in after the update the system complains that it was improperly shut down and whether it should report that.

Any idea how to debug this? The update worked just fine on the internal SSD without any issue.