Thank you for reaching out to Constant Contact API Developer Support.
I can confirm that, at least for the example given, there isn't an issue with the campaign_activity_id you're using - the referenced activity does have the primary_email role. The issue instead appears to be with the name of the property in your request body. The property name should be "scheduled_date", rather than "scheduled_date_time". If you still experience issues after modifying the property name, please let us know.
Regards,
Aarron G.
API Support Engineer
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