Your extractor fan needs to be connected via a triple pole isolator if it is the timer type or double pole for non timer & your bathroom light fittings need to have the correct IP rating //www.diynot.com/wiki/electrics:lighting:fan
If you compare 12 to 230 volt lamps at 1.5mm for 230v and 10mm for 12v and 50w lamps in both and say 10 meters of cable.
230v = 0.217A = 63mV volt drop
12v = 4.16A = 183mV volt drop
with 1.5mm cable 12v = 1.2V volt drop
Looking at those figures I can see why 10mm cable would be recommended however one has to consider how close the inverter or transformer is from the lamp!
If as in my bathroom a single unit supplies 4 lamps with separate cable to each lamp the 4mm would be ample but if daisy changed then the 10mm would make sense.
4 lamps equal spread on 10 meters of 10mm cable will mean last lamp has a 0.36 volt drop at 12 volt this is quite a lot.
Good reason for using GU10 where permitted instead.
10mm makes me think an error has been made - GU10 are 230v?
10mm is the size which you would normally use for an electric shower, bit on the big side for 5 mains 50w lamps!
Is the fan a timed variety?
Are you going to run the supply to the light switch, join the 2 neutrals in there, the 2 earths in there and run L, N & E to each lamp?
The circuit for the bathroom needs to be RCD protected.
This work is notifiable to your LABC under part p of the building regs.
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