Try to direct access the template URL page, using the below URL :
Using directly accessing the template URL page :
Using the PowerShell script :
$WebURL = "http://your-site-collection-URL"
#Get the Web Object
$Web= Get-SPWeb $WebURL
#Variables for Save site as template settings
$TemplateName ="PMO Site Template"
$TemplateTitle ="PMO Project Site Template"
$TemplateDescription ="Site template for PMO project management portal"
#Option to Save with content
$SaveWithContent= 1 #0 otherwise
#Save site as template programmatically with PowerShell
Source of the above script :
Save Site as Template in Sharepoint 2013 using PowerShell
Using the third-party migration tool (sharegate)
If that doesn't work, try to use the ShareGate migration tool, if you don't have, you can install the trial version of the product then do copy structure migration.
As far as server publishing feature is concerned - if you deactivate and re-activate it again - it should not create any problem. However, since this is in production server, it is not recommended to do it in production without performing this in test server. So, if you want to go with this approach, first do it in test server, then see whether this causes any issues - based on the result in test server take a call for production server.