Clicking and burning from the passenger side suggests the starter solenoid. I say that as I note that you do not have a starting problem yet, just an entry problem so do not worry about the starter for the time being.
I presume you have access to a voltmeter. I would be looking at the battery voltage right after engine shut down - maybe 12.6 volts DC, and prior to start up in the morning, should be 12.2 VDC or better. If you are seeing 11.9 or less, something is probably draining the battery overnight but not enough to inhibit startup. When the engine is idling and most electrical equipment off, the volt meter should show around 14 volts - maybe 13.7 to 14.2 VDC. Any less, maybe something is wrong with the alternator and the battery / system is not running at full strength. In cooler winter weather, the system may run about 14.2 VDC; in summer hot weather, 13.6 VDC. System voltage is ambient temperature dependent.
Also regardless of the voltage numbers, I would suggest you put some sort of charger on the battery overnight. I use a CTEK US3300 as a maintainer. It has a max amp rating of about 3.3 amps @ 14.4/14.7 volts. In my case, I will routinely use it all night for at least a week every few months just to bring the batteries back. I have two installed. The charging systems on our 3's just will not / cannot fully charge the battery regardless of how long you drive.
The reason I suggest the battery charging is that to find the problem, you first have to eliminate the simple stuff - a low battery is that. You may find with a full charge the door locks work OK, then later in the week, the problem starts again. That is progress - to be able to make the problem go away and then watch it reappear.
The key fob works because the roof mounted transmitter receiver hears your key fob signal and then tells the doors to unlock. A low vehicle battery could mean the receiver is hard of hearing. Since both your key fobs work the same way, I agree that the key fob batteries are probably OK, hence I start looking at the trucks battery system instead.
Also below I have provided the hard reset procedure instructions. I doubt the procedure will do any good but it does no harm. What it will do is erase some of the trouble codes that may be building up; I find it is a sure fix for when my radio starts to not want to make sound - display may show the station is there but no sound comes out - and then at a later restart, all is fine.
Hard Reset Instructions
In brief, to start the hard reset process, open the hood but close all doors and let your 3 go to sleep - no radio display etc and no key in the ignition. This will take a couple of minutes and do not open any doors until the reset is completed. (I find that it is often a good idea to have the drivers door window rolled down and the key in my pocket whenever fooling about the vehicle.) Also when I play with a car battery, I remove any rings I have from my fingers.
Disconnect the ground battery terminal from the main starting battery.
Disconnect the positive battery terminal from the main starting battery.
The reason for this order is if a wrench or loose battery cable ends grounds to the body, no arcing or other bad things happen.
(I assume that you only have one battery in your 3. If not, disconnect the others as well and leave them disconnected until after all is done.)
Connect the negative cable to the positive cable. (NOT the battery.) You will need a short length, (a foot or so), of light gauge, (14 to 18 gauge AWG, insulated stranded copper), wire to span between the battery cable ends as there is not enough slack in the positive and negative battery cables for the ends to touch each other.
Hold all together for about a minute or more, (at least two minutes), as you are discharging memory modules within the engine computer and elsewhere. Then it is suggested you then just let all sit disconnected for say ten minutes, (minimum five minutes), prior to commencing to put the battery cable ends back on the battery posts per the following order.
Reattach positive terminal to the main starting battery positive post.
Reattach negative terminal to the main starting battery negative post.
Start engine and hopefully no or fewer warning lights.
Reset time on the radio - note that the station presets are still there as not everything is erased. Nothing much else needs resetting either.
You can now connect up the other batteries if you have a multiple battery setup.
This is a link to a thread in disco3 re the hard reset procedure.
DISCO3.CO.UK - View topic - Hard Reset
This is a link to a thread on Disco3Club re the hard reset procedure.
Disco3Club The Discovery 3 and 4 Owners Club • View topic - How To, Hard Reset on a D3