The Aux Battery on the W205 should be a capacitor, and when an error message is displayed on the dash, this will need to be replaced. Replacing it without the warning message is unlikely to improve matters.
If your W205 is not starting/stopping, chances are your main battery will need to be replaced. But only replace the main battery after you have eliminated all other causes.
They seemed to have changed from traditional battery to capacitor in 2016/2017 it looks like so it might be a battery.
@OP - If it has the aux battery, I would charge it fully and see what condition it is in after that.
After 5 years, it might be shot but you won't know until you try. You should replace main and aux batteries at the same time really.
If it is a cap, I would replace it personally as I've read they can be troublesome.
This was my previous post on it - with how to find the aux battery if it does have one.
Where is auxiliary battery in w213 E220d located? | Parts, Maintenance & Servicing
This will show you where the battery is and you just need access to the terminals to charge it like you would imagine.
I wired in a couple of wires on to my aux battery so I can charge it through the fuse hatch at the rear.
If your aux battery has never been charged, it may take a while to fully charge but it's a small battery so we are talking an hour or two at a guess.
My aux battery gets a charge once every quarter now and takes a few moments to fully charge now (and also it was new a year ago).
That will sort the stop/start.