SSL has absolutely nothing to do with Local Publishing.
You can do Local Publishing With SSL, which requires *two* certificates: a CodeSigning Certificate for publishing and a WebSite Certificate for SSL
Or you can do Local Publishing without SSL, which requires only the CodeSigning Certificate, and that cert never gets anywhere near IIS.
Not entirely sure what you're doing, but I've been involved with thousands of Patch Manager local publishing installations, and done a few dozen of my own, and I've *never* had to do anything with SSL.
Now.. if you chose to enable SSL on your WSUS site (not required, but certainly an option), then of course you need an SSL certificate for that -- but it has *nothing* to do with publishing.