I'm thinking your trailer battery should also charge through the trailer wire connection. Back up, plug up and idle the tow vehicle for a while with a nice cup of coffee then drop the hitch down and connect the rest.
A dead battery will also prevent emergency trailer brakes from functioning.
If your battery drains like this is probably shot, it's time to install a battery cut off breaker switch right at the positive battery post. Example pictured, but you probably want a 30 amp. My batteries do the same, but I added solar and they're always up to snuff now.