Hi,
I've been experimenting with this for some time now.
You can charge the leasure battery using a batter charger plugged into an inverter that is running off the car battery. No problem.
I had the idea some time ago of hooking the 'van upto the inverter. The idea being that I could top the leasure battery up each time I visit the 'van for maintenance(pottering about). Also, it would be a good way of filling up the battery from the car when without mains electrics. The car battery shouldn't get too low as the inverter will cut off when the donor battery voltage starts to drop.
I'm not sure on how the zig unit works though. I hooked the 'van to the inverter and turned on the zig unit along with a few lights. I expected the inverter to come under load, but nothing happened. The battery was in good condition and took little from the charger.
I'm not sure, but I think the zig unit takes very little mains wattage regardless of the demand for 12 volt amps, any deficit (which happens when even the smallest light is turned on) is taken from the battery, this is then replaced at trickle rate at night, when all is turned off.
The short answer is "yes". If you're looking to quickly top up, I'd use a charger though and not the zig unit.
Also, note the charger should also be a trickle charger and not a booster thingie as they tend to bend the plates on leasure batteries.