The son must sign with the Draft. There is no exemption for a declared war, although there are some accomodations made.
The survivor rules you refer to change from time to time. But basically to qualify for a survivor status, the son, the person must be one whose parent or brother/sister was killed in action, died in line of duty
, or died later as a result of disease or injury incurred in line of duty while serving in the armed forces
of the United States; or have a permanent 100 percent Service-related disability (including 100 percent mental disability), as determined by the Department of Veterans Affairs or one of the Military Services, and are not gainfully employed because of the disability. Also included are those whose parent or brother/sister is in a captured or missing status as a result of service in the armed forces during any period of time.
The rule does not apply when Congress has declared war (it has not for the current deployments to Iraq or Afghanistan).
Here is a link to the primary DoD
regulation applicable. You will see the peacetime exemption could apply.