Currently, Salesforce Chat Transcript Highlight Panel is not customizable. When the Chat is Active, it will display the first message. When the Chat is completed, it will display the Chat Transcript Name which is an Auto-Number field.

Salesforce Idea:

https://ideas.salesforce.com/s/idea/a0B8W00000GdayEUAR/change-the-highlights-panel-on-the-live-agent-transcript-lightning-console-page

Salesforce Known Issue:

https://trailblazer.salesforce.com/issues_view?id=a1p30000000T3hNAAS

Workaround using Related Record Component:

1. Created or Edit the Lightning Record Page used for Chat Transcript object.

2. Remove the Highlights Panel component.

3. Add the Related Record component.

4. Use New Update Action to create a new Quick Action.

5. To adjust the fields, use Edit Layout button on the Quick Action.

Output: