[Johnnyroper]

So battery has lower but there’s no power at fuse box? If that’s the case you need to check connections on the power distribution that lives on top of battery to make sure everything is clean and tight. Also while at it check the earth connection from battery to body is good

 

[Johnnyroper]

Been getting warm for a while by looks of it, also that knurled nut doesn’t look standard to me either.

 

[Johnnyroper]

https://www.realoem.com/bmw/enUS/showparts?id=3D32-EUR-03-2013-F30-BMW-320d&diagId=61_2891

I would get a new BST cable if it was me number 1 in above link, the BST trigger that’s melted in I would cut one off a car in scrap yard and splice it in to yours. You wouldn’t specifically need f30 one as most Bmw have same plug. Just get enough length to cut away damaged bit on yours.

 

[Johnnyroper]

Yes it would cause it but you also need to establish what caused the problem, could just be poor connection at battery which caused heat but you want to make sure there’s not an underlying problem.
Absolutely no idea what they would charge, I would be replacing the battery aswell if it was mine.