We cannot add the complete full Static Resource URL in the Embedded Service Branding. If you add, it will show a broken URL.

Reference Article:
https://help.salesforce.com/s/articleView?id=000336442&type=1

To resolve the issue, the relative path should be used in the Embedded Service Deployment Configuration in the Experience Site Builder rather than the Full Static Resource URL.   

Syntax:

/{Experience Cloud Site URL Path}/resource/{Static Resource Time of SystemModStamp}/{Static Resource Developer or API Name}

Note:

1. Here cc refers to my Experience Cloud site URL path.

2. Following SOQL can be used to get the Static Resource Time of SystemModStamp.

StaticResource objSR = [
    SELECT Id, Name, CacheControl, SystemModStamp
    FROM StaticResource
    WHERE Name = 'Avatar'
][ 0 ];

System.debug( objSR.SystemModStamp.getTime() );

Output: