No, activating the support licenses will not affect whether a extension or template will function or not. That is only used to determine if it is eligible for in admin updates.
The issue, that really depends on what is not working. Likely the cause is in the method that was used to move the site from the dev to the production servers or the misconfiguration of some of the WordPress constants in the
I’ll need more details in order to better assist. For example what and where does not work and what, if any errors are you receiving.
If you could open the site to me that will likely be very helpful and make resolving the issue as fast as possible.