...does the job date from the original CL carry thru as the job date to the CL called in the SBMJOB command?
Generally, no. A submitted job's JobDate is normally the value from the QDATE system value at the time that the submitted job starts.
But technically, it's not possible to be certain just from your example SBMJOB command.
Any number of commands before or after the SBMJOB command could have modified the QDATE system value temporarily. Or the SBMJOB DATE() parameter might have either *JOBD or some valid date set as a default. For *JOBD, the DATE() parameter on it could be set to some unexpected value.
Or there could be exit programs registered on your system that affect how SBMJOB parms are handled. Or there could be a routing program that modifies a JobDate. Or...
In short, it will probably be the system date when the job starts. It just can't be guaranteed without more details being checked.