First thing, Id read up on the switch you have and make sure you can switch it while the motor is running, on some or them you cant without risk of damage. Since both batteries are the same type combining them would be fine but yes it would also bring the good battery down by combining with the dead one.
It just makes more sense to me running everything off of the both or common post and I believe this is the way the switch was meant to be used. One other thing. Your alternator isnt really meant to be used to charge your dead battery but really to just maintain your batteries.