ros::time corresponding to message sent

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No, afaik the time a message was published is only available if the sender has added that to the message (and even then it's an approximation, as they will probably use ros::Time::now().

The MessageEvent class can provide you with some additional information, such as when a message was received (by your node), but not when it was sent unfortunately.

Also see wiki/ROS/Connection Header for some more fields that can be accessed that way.

I'm doing some clock synchronization with another package with custom messages that don't include a ros time in the message definition.

To synchronise data between nodes a Header (or similar structure) is required, and I would say, is actually exactly what it was intended for. But you probably already knew that.

EDIT: The ros time of receipt within a rosbag would also be a help, if there's a way to recover it.

See #q199941.