By default when you click the Live Chat link in the header of the CSM Service Portal it opens the virtual agent widget. This explains how you could switch this to use Connect Support instead.
Please note that this process is not something which is built into the platform and requires either switching or editing widgets, which is a customization. Any further issues this causes would be considered implementation related and not supported by Customer SUpport.
To make the Live Chat link open Connect Support in the CSM portal you need to switch or edit the Header widget referenced in the Theme of the CSM Portal to the Stock Header or a custom header widget which uses the code form both the CSM Stock Header and the Stock Header.
Relative URL to the CSM Stock Header widget.
Reliative URL to the Stock Header widget.
You can either replace the 'CSM Stock Header' referenced in the theme of the portal record with the 'Stock Header' or edit the code in the CSM Stock Header to use similar code to the 'Stock Header'. To get to the reference which is currently pointing at the 'CSM Stock Header' please follow these steps.
1. Navigate to Service Portal > Portals.
2. Open the record for the csm portal.
3. Click the reference on the Theme field
4. That record should have a Header field which is referencing the CSM Stock Header.