can you look behind the shelf ? see if there's any water damage on that part of the wall.

and when you say the leak only happens when using the tap and sink, did you try just using 1 to be sure ?

a possible scenario is that your tap pipe has a leak, if it's GI pipe with built up rust inside water needs abit of pressure to come out (rest of the time just "moist" on the surface). so when you on your tap and the water pump kicks in you get a wet wall.

you could try to remove the tap and see if you can fix it from toilet side ?

if that dosen't work and you still don't want to hack then run a surface water pipe to where your existing tap is. Ugly but no hacking