Background: I purchased my 2012 XKR a little over a year ago. It was a trade in for a new F-type . It had a battery replacement by the dealer due to lack of use as I bought it with 6000 miles on it. It sat for a month prior to it being shipped to me. When I got the car the 1st thing I did was put my 10 amp charger on it until the battery level was normal. I then keep my Sears smart charger on every time I park the car and during last winter. I had also purchased a plug in volt meter which is plugged into the accessory port in the glove box.
Every time I use the car I unplug the charger and then open the door which energizes the volt meter. The voltage is ALWAYS 12.5 V on the meter. When I start the car you see a quick drop in volts to 11 but when fired it goes to 13.5V and stays there all during the drive whether day or night or hot or cold. When I shut it off it goes to 13V and down to 12.5 in less then a minute at which time I lock the car and attach the charger. It has been this way for the last year.
Don't know if anyone is interested but just wanted to pass on my experience.

 

I just put a new battery in my XK at the dealer, and they "replaced the battery monitor module, (negative battery cable)" and Also "performed test of the BMS supply wire white and blue". When I got home I opened the hatch and plugged in a Battery Tender LCD voltage indicator to the harness that goes to the remote jump-start connections. It registered 12.4 volts. I did not shut the hatch or lock the car; therefore the two hatch courtesy lights stayed on as well as some of the other modules that stay on for about 30 minutes including the fan? behind the Nav screen. When I came back 1 hr., later everything had shut down and the voltage had jumped to 12.6 and settled there. I put the Battery Tender on and the next morning the battery showed 13.1 volts. Took it on a drive and made a couple of short stops and when I came back to the house, opened the hatch and got an immediate reading of 12.6 volts. I came back an hour later and saw it had bounced to 12.8 where it stayed for 3 days. I took it back to the dealer because I thought that 12.4 & 12.6 was a little low for a brand new battery. They kept it several days and said they thought this was normal after talking to the Jag hotline. It just makes you wonder how low it could go if you did a lot of stop and go driving with a few stops along the way and did not lock the car every-time to shut all of the systems down or carried the key fob around while cleaning the interior/exterior etc. in the driveway.

I've never had a voltage indicator on this car until just recently so I'm kind of **** about checking it. The dealer asked me if I had ever gotten a warning message indicating a fault about the battery having a low charge or not charging. I never knew there was any such message that could appear and have read the section about the battery in the owners manual. I've never seen that message appear. Hope this info is of some benefit to you

 

Somehow I am not comfortable with what you are experiencing. I would get a volt meter that plugs into the accessory port in the glove box. Read the voltage when the car is running. With a fully charged battery you should see between 13-14 volts. I used something like this 12V 24V LCD Digital Cigarette Lighter Car Voltmeter Gauge Electric Voltage Meter | eBay

 

The battery warning comes up if you leave things switched on & don't start the engine - for example, if you sit in the car and keep the audio system playing. It doesn't necessarily mean the battery is bad, it's just telling you you ought to start the engine to avoid running the battery flat.

If you want to see how the voltage goes when the car is locked without having to position the meter so you can read it with the car closed, you can 'fool' the locking system by opening the hatch and then flicking the latch to the closed position using the shaft of a screwdriver. You can then lock the car using one press on the remote, and it will go into quiescent mode much faster. It's also a useful trick if you want to do some work in the rear without flattening the battery - make sure you unlock the car again before closing the hatch.