Patching FileShareWitness for AlwaysOn AG
Normally, rebooting the File Share Witness should be totally fine in this setup as long as both AG replicas are up and healthy when you patch the FSW server.  So you shouldn't need to remove anything from the WSFC settings to accomplish your plan.

I had a situation a while back where


the file share witness was down for maintenance
the PRIMARY and SECONDARY both switched to the RESOLVING state briefly (about 15 seconds)
then they both returned to their original roles  


This was due to the File Share Witness resource running in the same Resource Hosting Subsystem (RHS) process as the AG node resources - when the FSW failed it's "are you up" checks, it caused the whole RHS process to fail, which resulted in the "blip" in the AG.

You can avoid this problem by setting the FSW resource to run in a separate RHS process in WSFC settings:



I blogged in detail about that here: Troubleshooting an AG Failure



If there is something else causing this problem in your case, you can still troubleshoot it using the steps outlined in my blog post:


look at the SQL Server error log
look at the WSFC cluster log
review the AlwaysOn_health extended event session


If you don't want to dig through every single one of these places, you can also use the Failover Detection Utility provided by Microsoft Support.