I need to install Java for Mac OS 2017-001

If you did not perform a last Time Machine backup of the previous macOS before upgrading to Catalina, or if you used the same Time Machine drive for a first Catalina backup, then you cannot go back to the prior operating system without a clean install.

If that Davis Weather app uses Java 6, or if it doesn't, what does Davis have to say about the future status of their Weather Link software?

Since you have an investment in that local weather monitoring setup, and if you do have a prior Time Machine backup, then you can shutdown Catalina, connect the prior operating system Time Machine drive, and then boot into Recovery ⌘-R and select Restore from Time Machine Backup. This will wipe the Catalina installation and reinstall the prior operating system and all of your files as if you had not installed Catalina.

Apple's Java 6 implementation that you refer to is dead on Catalina, as there are no longer any 32-bit applications supported that originally required that old Java version.