No, there is not a way to do what you are looking for.
I take it that your multiple catalogs are not grouped near each other in a "LR_catalogs" directory tree?
menu:/File/open recent/ will show the name of your current catalog (and the others). The path will appear for any catalogs with duplicate names.
You might be able to edit the "identity plate" in the upper left corner with the path if you wanted to.
The name of the catalog (but not its path) is displayed in the title bar of the main window.
If you are in Normal window mode (i.e. you have a window with the name of the catalog file) and you are on Mac OS, you can just right (or control if you haven't set up your mouse or trackpad to be two button) click on the name and you'll see the path. I believe windows can do something similar but I have no way of checking that.
Neat, I didn't know that.
On Windows, if you right-click the title bar, you then choose the option Show Catalog Location -- not much different than Edit > Catalog Settings.
If you hover your mouse pointer over the name in the title bar, on Windows LR will give you a tooltip window with the path.
This and the right-click on the title bar is new to me, too.
Thanks! That's perfect. Close enough for daily work.