Couldn't you re-register your old phone temporarily, just for the purpose of taking the backup¹, and later re-register your current phone? (In fact, once your old phone is registered, turn on airplane mode, and you should be able to re-register your current phone right away without affecting your old phone and the backup you're making there.)

Alternatively, would it be an option to get a throwaway number you could register your old phone under?

Finally, once you have the backup, use something like https://github.com/bepaald/signalbackup-tools to merge your old phone's backup with your current phone's backup, and then reinstall Signal on your current phone from that merged backup. (Disclaimer: I have never actually done this before but signalbackup-tools has been around for a long time and the developer seems to be very responsive.)

¹) I'm talking about the traditional way of backing up Signal conversation data to an encrypted archive here, not the feature discussed in the OP.