Hi Laura,
The camera produces the same amount of heat in live preview as it does recording, since it's always running the image sensor at full speed, so there's no thermal benefit to having the camera on but stopped. The way to do a start trigger is to use a stop trigger with a delay equal to the record time, and leaving the camera recording just like post trigger.
The camera does get quite warm in a waterhousing, but it's not a problem. Do keep in mind you need to do a black cal while the camera is at full temperature to get the best image quality.
David