You can use Dosdue1’s Catalina patches to download the official Catalina installer from Apple’s servers.
Disclaimer: I’m not a fan of bodging macOS to work on unsupported platforms, especially for “production” machines. However, if you want to tamper and this is something to experiment with learning, I say have it. If you need the latest operating system for work, get a machine that supports it.
Basically, what you need to do is trick macOS and ultimately Apple’s servers into thinking that your computer is capable of running Catalina so that it will download.
When running Patches, clear the “Apply automatic installation updates by mail”
When you reach a screen where you can browse an installer or download a copy, select the download option.
Download DMG, it will be about 7 GB. From there you can create your offline installer. The terminal command is as follows:
sudo /Applications/Install macOS Catalina.app/Contents/Resources/createinstallmedia --volume /Volumes/Catalina --nointeraction
If this works out for you, I encourage you to buy him a coffee or a beer (or maybe even dinner) by sending a donation in his own way (I’m not affiliated, I just appreciate people’s hard work) . You will find the link “donate” on the software download page.