Just so I understand: You have a bootable macOS installer with the latest Catalina version 10.15.7, patched by Dosdude1's Catalina patcher. Is that correct?
If you have the macOS installer mounted on your Mac, then open System Preferences/Startup Disk pane (unlock that pane now), you should see your bootable installer as one of the selections. Good so far? If you mouse across the boot choices, you should see a macOS version number AND a build number.
What version and build number do you see for your patched bootable installer?
Thanks. I'm not at the house now, but the answer to your first question is Yes.
I will do the Startup Disk pane when I am back there. But I can tell you that under the About this Mac System Report it says the build number is 19A603.