Well, you made the right decision. If I were to migrate to Catalina, I would proceed as follows:
1) On an external drive, create two logical volumes to create bootable backups on. Name one Mojave and the other Catalina.
2) Use a product such as SuperDuper running on your Mojave system to create the bootable Mojave volume on the external hard drive.
3) Test that you can in fact boot Mojave from the external hard drive. If you cannot boot from this drive, do not continue. If you can boot from this drive, you now have a bootable, restorable point-in-time (PIT) backup of Mojave.
4) Now do a clean install of Catalina on the INTERNAL hard drive (overwriting your original Mojave system: this sounds scary, but you have already demonstrated you can boot from Mojave off the external disk if you got to this step, so you've lost nothing). Make sure you apply all outstanding Catalina updates.
5) Reboot Mojave from the external hard drive. Using SuperDuper, create a bootable Catalina backup on the Catalina volume on the external hard drive. When this is complete, you now have PIT backups of your original Mojave system and the new Catalina system you are building.
6) Test that you can in fact boot Catalina from the external hard drive. If you can, you now have a bootable, restorable PIT copy of a clean Catalina install with all maintenance updates applied.
7) Using SuperDuper, restore your Mojave volume from the external drive to the internal drive, overwriting the Catalina installation. This is OK because you have a known bootable working version of Catalina on the external hard drive. This step puts your system back the way it was when you started this process, so when you boot it up, it will come up running Mojave.
This method allows you to boot up your Mojave system off your internal hard drive so you can use your computer as you always have, and when you want to work on beating Catalina into submission, simply boot off the Catalina volume on the external hard drive. Now you can install your apps, copy over files and get help with troubleshooting issues at your leisure.
I do my upgrades this way because I don't think the OS X installer will let you install to an external drive. I could be wrong about this, and if I am, then you can just replace steps 2-7 with the install of Catalina on the external hard drive.
Note: I am retired from a major disaster-recovery provider, so if these steps seem overly anal, that's probably why. I live by two maxims:
1) Always have a current bootable backup.
2) Always be able to put your system back the way it was if you make changes to it --- which means, take a bootable backup BEFORE you apply maintenance updates to either Mojave or Catalina.