Here's excerpts from a magazine tech tip that I saved a few years ago. I don't know if it does apply to the C5, but it does to many late model vehicles. To be safe I never jump start with the charger on, or from a running vehicle. Better safe than sorry.
"When I went out to start the car, it wouldn't. All it did was click. I got out my trusty Sears battery charger and hooked it up to start the car. When I did, it was backfiring, stalling and hesitating. I took it to the Chevy dealership and they told me I had "smoked" the ECM by hooking up the battery charger. My question is, can jump-starting a Corvette damage the ECM or are they just giving me a line?
Answer: Your (C4) has a normal 20 milliampere bailey draw. This low battery drain is due to the various microprocessors (computers) in your Corvette. If the key is left in the ignition there will be an extra 180 milliamp draw due to the fact the central control module (CCM) is then awake.
You should not quick-charge a Corvette battery. A slow 24-hour charge is better. You can "smoke" the ECM by putting a 50-amp charge on a low bailey. When jump starting a car, use the normal 12-volt battery-to-battery method. When you turned on the key with the 50-amp charger connected, the ECM received a high spiked voltage. When jump starting, hook up cables then let it sit for a couple of minutes before tying to start it. This powers up all of the microprocessor boards and prevents damage.
If you start the car with a charger hooked up, the charger sees a low battery condition and kicks in. This voltage spike can damage electrical components."