Automate the patching of SQL Severs on Windows
While the rotating passwords are a hindrance, they don't negate any approaches available.  Kendra Little does a great job of explaining a general approach to patching servers in an automated fashion in this blog post, as well as some of the risks about when server patching goes bad.
Because your servers are not production, these would fall under her &quot;cattle&quot; classification which means that if you can tap into an existing automated toolset such as SCCM/WSUS (as denoted by Alen prior), you should do so.  Patches that don't go through may be annoying, but they hopefully won't cause outages that wake you in the night.  The rotating password complexity may require you customize this to a degree, and really the best approach there is to automate the process as best you can using PowerShell scripts.
Adam Bertram probably has the most robust toolset out there now that I'm aware of in this domain.  I would suggest you check it out and see if you can augment it to your needs.  Here's a direct link to his GitHub repository for the same toolset.