SQL Server - How to restore the database except specific tables
To exclude certain tables from your refresh of the integration environment, with a backup from production, you will need to make use of a staging\data cleansing environment.

The plan would be:


Backup production
Restore backup in staging 
Drop the unwanted tables
Backup staging
Restore backup in integration


This plan could easily be implemented with PowerShell (DBATools is well worth a look) or SSIS. 

If you like coding, go for PowerShell. If you like dragging icons around the screen, go for SSIS. Both can get the job done.