Welcome.  The leisure battery will charge from mains hook up and from the car's 12V system IF your towing electrics include that function (it's not always included as standard).

Automatic brake adjustment does not sound like ATC to me.

If you already have a caravan, I wouldn't think you will urgently need any extras.  Better to wait and see what might be useful when you've used the Eriba a couple of times.  As you know the 410GT doesn't have a toilet/washroom - would you want a PortaPotti?  We don't bother.  Most people also want an awning - again we don't bother.

Welcome Nelson,
as above the battery pack, sometimes called Autonomy, should include a battery and the relevant charging when on hook up. Your car if set up for it should also charge the battery when on the road along with provide power to the fridge. This is not always the case on VAG cars so you may need to check.
The automatic brake adjustment is just that, as brake shoes wear an adjuster mechanism does it's job. It is not a form of stability control such as ATC.
Regarding a portable loo for your new 410, having had a portable Fiamma for our loo-less 310 I would advise getting the tallest you can store.
As a current caravan owner you will have the majority if not all of the gear you need. The only exception is probably an awning. You will know if you need one from past experiences. If you do then new Eriba fit awnings are fairly limited. A Kampa Pop-Air Pro 340 would meet your needs if you require something suitable for 2 but not a family. It is fairly quick to put up, easy/quick to take down and reasonable value new. You must however make sure you get one of the 2019 onwards models for your van.
The Isabella Shadow is a common purchase for sun protection and will take fairly heavy rain. You will need the 3m version.

Colin

A thought regarding ATC since you asked about it - many recent cars have a trailer stability program, a modified algorithm of the car's electronic stability control, which is active when trailer electrics are connected.  Not sure if AlKo's ATC can be fitted to Eribas and I can't think I've heard to anyone doing it.