Well, the low tire pressure can be blamed for maybe up to 5 miles of range lost. The temp being below freezing reduces the advantage of the heat pump, which loses most of that advantage at about 25F, and virtually all of it at 15F. You can also assume that the 12th bar, worth 15% of capacity, is soon to drop. (You really need LeafSpy Pro and a Bluetooth OBDII port reader like the one in my signature to read capacity accurately - although LeafSpy is also fooled by a BMS reset.) So assuming 85% of capacity left, and a 10-15 mile range disadvantage for the above reasons, you should have had about a 75-80 miles range with a full charge. If the range really was 53, then we are likely looking at a BMS reset, unless the test drive was more vigorous than you imply. Actually, any distance driven at 75MPH will produce a fairly large range hit...
The dealer mentioning the 12 volt battery having died may or may not be significant. That doesn't generally cause a BMS reset, but I wonder if a stone-dead 12 volt battery for a few days might have that effect, or if the dealer reset the BMS and is trying to cover their tracks. You'd really need to do a better range test on the car - tires to 40psi, no faster than 60-65MPH, temps no colder than freezing, to get a better handle on the issue. Start with a full charge, drive 10 or 20 miles normally, then note the drop of state of charge on the dash display. Or drive until you lose 10% of charge, and multiply the number of miles that you traveled by 10 for a rough range estimate.
Brilliant Silver 2021 Leaf SV40 W/ Pro Pilot & Protection
2009 Vectrix VX-1 W/18 Leaf modules, & 2 lithium E-bicycles.
BAFX OBDII Dongle
PLEASE don't PM me with Leaf questions. Just post in the topic that seems most appropriate.