Would putting a second battery in my trunk put more strain on my already struggling 80amp alternator?

Actually no. Your alternator provides the power necessary for all the equipment in your car. For example turning on the radio uses more alternator output. A cigarette lighter draws major power but only for a short period of time. Using the lighter to charge your laptop would be a high draw for a longer period. Having two batteries does not require more power than one IF both are charged. You could consider both as the equivalent of one huge battery. If you left your lights on by accident and drained BOTH batteries your alternator would work harder to charge both.