As you say "dumb" but not the first
plus your battery sounds scrap any way 396/600 CCA.
Most of us including mechanics, unless we read up about it in a trade journal etc. would never have come across needing a "specific " battery to start a car. Having spoken to auto electrician regarding this, the batteries , starters and alternators are more expensive than their predessors and do not last any longer, all in the name of protecting the environment, but unless you live in the city, having a car that stops at every junction is a pain in the ar*e. as far as I am concerned.
It seems the drop in battery voltage due to incorrect battery type during cranking especially in colder weather can aggravate the problem.
Many on Forum have written re this issue, but as I understand it if you have Stop/Start function etc. then you must have the correct battery, also there is a procedure when changing those batteries which if not followed can cause similar problems even with the correct battery.
I am assuming the rest of your charging system is in perfect working order, as any low battery voltage can also cause further related problems.
So basically as I understand it, the correct battery, correctly fitted, assuming charging system all good is required , however you will also need a good diagnostic tool (many here use
MES with success, I have used Snap On Zeus) to delete any system error codes and then perform "proxi alignment" to tell BCM etc. what is connected to the system.
One other side issue, many have related problems due to poor earthing, so it might be wise to check that also.