Whether through official or unofficial methods, what way can I build projects using an older SDK, say version 10.12, on Xcode 13?
On Xcode 12, I was able to paste an older SDK into /Applications/Xcode.app/Contents/Developer/Platforms/MacOSX.platform/Developer/SDKs/
However, that appears to have no affect on Xcode 13.
See here for agx's answer to this.
What problem are you trying to solve? There are usually other solutions rather than changing SDK versions. Using an SDK version other than the one shipped with Xcode is an unsupported configuration.
I am trying to maintain a KEXT as we still provide support for users on Catalina, but the code will not compile under the latest SDK as it does not contain many essential functions and structs from headers inside <security>.
In that case, you need to download the Xcode version with the SDK you need, and run it on the associated macOS version. Since you mention the macOS 10.12 SDK, that's Xcode 8 running on macOS 10.12.
Unfortunately, that is the situation I am trying to avoid. I'm putting it lightly when I say that it's unfortunate that we have to go to such lengths to develop older SDKs. Appreciate the input, though!