You do have a leisure battery there by the sound of things, and that's quite a good make. If I were you, I would get a battery charger, take your battery home, and put it on charge. It is probably quite flat. Hopefully, if it has not been flat for too long, you can recover it by simply charging it up. If it won't hold a charge, you will need to replace it unfortunately.
Charging from the car while driving would take simply ages, if it charged at all. If it's really flat it may not, because the charge from the car is at a really low level, really intended to keep a battery charged rather than charge a flat one.
What they call a smart charger is what everyone on here usually recommends, but I just use an ordinary battery charger that I have had for years. If you use an ordinary charger like mine, you need to put the battery on a high charge for a while to boost it back up. Then switch to a trickle charge. I think the much newer smart chargers do this automatically. Someone on here will advise about which one, I'm sure.
The second 7 pin socket is mainly for powering the fridge, as you say, although it does provide a charging supply. My car isn't fitted with the second socket, and it hasn't caused me any problems so I haven't bothered fitting one. Last year I drove home from a holiday in Cornwall, a 300 mile drive, and my fridge was still cool enough when I got home. I had been running it on the mains onsite.
My battery in the caravan still has a decent charge left in it, and it's been outside in my caravan for at least a couple of months, and it's only a car battery not a leisure one. I really must remember to go out and give it a top-up charge. All the lights still come on bright when I go out to the caravan, and the green light is still on. It goes out when the battery starts to run down.
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Topic: Do I have a battery?
