All interesting stuff re stop/start. Had to change my aux battery 18 months ago but the stop start rarely kicks in unless I drive 150 miles or more.
I suspected the main batt may be the cause but while it still starts the car I think I'll live with it..
Probably is the case as there are a few conditions that have to be met in order for the car to be ready to STOP/START.
Certainly the state of both of your batteries has an impact even though the aux one is used for STOP/START.
Having had a new aux battery before I bought my car, and then running on the original battery still, it took a long time for STOP/START to be ready.
Then that battery was replaced but the new one turned out to be a duffer, but that's another story.
But anyway, MB then replaced both batteries under warranty weirdly, even though I only expected them to replace the main battery.
After that, it was STOP/START ready a lot more quickly.
Then Covid happened and I stopped using the car daily but since then, I trickle-charge my main battery when not using the car and charge my aux battery once every 3 months.
The car is always ready to STOP/START very quickly from starting no matter what time of year now.
All the more complicated E and S Class cars (has been the same on my W211 E55, C216 CL and now W222 S500L) are seemingly very fussy about the state of battery charge.