Honestly it looks to me as well that their advice is solid as far as what to do and why.
I'm smacking my forehead on #5, I hadn't thought about a relay or anything the battery and cig lighter. All my cars have always had "live" cig lighters at all times but if the 1 series cuts the connection then, yep, not a good way to charge the battery through.
And though I was suggesting the cheaper charger, if the BMW model guarantees future work if needed then it was money well spent. in the end I think you have done well.
Also FWIW sounds like you've found a good shop, definitely go back to them if / when needed.
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