What version of Pages will work on Catalina10.15.7... and how do I get it and keep it?

Pages/Numbers/Keynote v11.1 was the last version release of these applications for Catalina. If you ever previously installed these applications since Oct 2013 from the Mac App Store, then you may be able to retrieve these versions for Catalina. If you never installed them, or if older versions were pre-installed by Apple, then the suggestion to use Pages for iCloud is your only option.

If you did previously install these applications, launch the Mac App Store on Catalina and sign in with the identical Apple ID that you used to install them earlier. Next, press cmd+0 (zero) to open the Purchased panel and locate these applications. If there is an update, install, or a blue cloud icon present, click it and a dialog will appear offering to install the last compatible version of these applications for Catalina.

http://www.icloud.com/ gives you a web version of Pages you can use.

The App Store might give you a stand alone as part of iWork. If it doesn't LibreOffice, OpenOffice, OnlyOffice are all great stand alone alternatives.

I never suggested upgrading. Those other free tools are alternatives that can be used in Catalina if for some reason the one source doesn't work.

Catalina was the first 64 bit only operating system version of Mac OS X. This meant older applications written for the Mac that weren't 64 bit will not run on it.