This has been driving me nuts trying to find information on this, so sorry if this is a re-post, but a search for 'krita file name' brings up a ridiculous amount of results, and I can't find it in them.

Whenever I save a .kra document with 'Save As' and try to save it under a different name, it will always display in Krita as: "Old file name - new file name" whenever I open the file.
Is there any way to change this?

This is on Windows, and regardless of the actual file name in the folder, Krita keeps the old name attached.

Go to the file/document information dialog and change the title field.