Calculating wire loss is simple, here's a US version with AWG. I'll use metric, so 1.5mm2 wire is 11 mOhm/m.
One could say that higher voltage allows using a thinner cheaper wire, however building code and economy interferes with the calculation. For example here, 1.5mm2 wire is the standard for lights, so that would be used in the 240V case. Thus the 12V case is more about using the same 1.5mm2 versus thicker wire.
Another complication is that 230V LED lights do not have power factor correction, so the current they draw will not be a sine, they would draw high current peaks of unknown amplitude and duration. This also depends on the waveform at the output of the inverter. If it's a square or modified sine, the cable losses will be different...
However cable losses will still be rather tiny, let's consider a 10W bulb, on 230V this would draw 50mA RMS (rounding up), let's fudge this to 150mA peaks with a duty cycle of 1/3, this gives 2mW losses on a 10m wire, which is negligible.
Thus let's neglect the cable losses for 230V.
For 12V the same bulb would draw 0.83A DC which amounts to 76 mW per 10m of wire. Again not too bad, but remember losses are in RI^2 so they are proportional to the square of the number of bulbs...
Also losses are in RI^2 but R is inversely proportional to the copper cross section which determines the cost of the wire. A wire that is twice as expensive (2x the amount of copper) will not halve the losses! It will only reduce them by 30%... (square root of 2)
Anyway. Draw your installation with number of bulbs and wire lengths.
Compare the lumen/watt efficiency of 12V and 230V bulbs. Also the price and availability. Compute how many watts/bulbs you need for your target lumens. You should of course compare two installations with equal light output...
So you get the 12V current, and thus the cable losses. Most likely they will be tolerable.
Now add the inverter losses to the 240V case, looking at the inverter datasheet, and compute how much current each case will use from the 12V battery...
My guess: the winner will be determined by the efficiency of the 12V LEDs you find. Remember efficiency is not the only criteria, color rendition is also very important for comfort, and good color rendition increases cost and decreases efficiency.
Make sure you get a sample of the LEDs first before buying the whole lot, and make really sure they look good to you. Some have very high efficiency, but they look nasty! (like: greenish, bluish, make people look like aliens, make your blue veins stand out, etc). These are for the corridors only!
I put CRI95 LEDs in my kitchen... the food looks good ;) some LEDs make your fresh steak look like roadkill from a month ago...
