Thanks for asking question! If I understood right the updates fail with: Unable to add XXXX.dll to the Web site when trying to publish to an Azure Linux App Service. If so to fix this add the appOffline=true to the ARM template or use the AppOffline switch to the MSDeploy.exe command line.
Also, another option you may try is to rename locked files.
Set MSDEPLOY_RENAME_LOCKED_FILES=1 to Azure App Setting on your app. This causes msdeploy to attempt to rename DLLs if they can't be copied during deployment. This often works because even when DLLs are loaded, they can typically still be renamed. It renames them with a .delete extension, which it then cleans up on the next round.
You may refer to below link for more details on this:
https://github.com/projectkudu/kudu/wiki/Dealing-with-locked-files-during-deployment