https://github.com/ps2dev/ps2sdk
Maybe this SDK will meet the expectations? Of course, I haven't used it, but it's open source and there are some tutorials on how to compile it.
If it does not have to be an open source SDK, maybe the 2005 version available on Internetarchive will also be of interest.
https://archive.org/details/PlayStation2July2005SDKversion3.0.3