The trouble is, because of the position of the shed, any bamboo you plant either side of the shed door isn't going to screen that balcony efficiently; if you could relocate the shed to the left hand side, a small tree such as Malus 'Gorgeous' judiciously placed 1 or 2 metres from the right hand fence so that the crown will exclude the view from that balcony would work well. Yes, it will reduce sunlight towards the end of the day, but so will anything of any height placed there if it's high enough to block the view, and at least it's only to the right hand side and will allow dappled sunlight through.
If it's that balcony you particularly wanted to block out, other methods of screening aren't really viable; trellis sections added to the back fence would block out the view from the other windows which are lower down, but obviously, not the balcony. If it's not possible to relocate the shed, then planting a small tree three feet in front of the shed, to the left of the door, would block the balcony view - but would also block out more sunlight, unfortunately.
If you want to try with bamboo, Fargesia nitida https://www.rhs.org.uk/Plants/63525/Fargesia-nitida/Details is the one to go for - plant at least two feet in front of the shed, to the left of the door; when it gets tall enough (ultimate height 2.5 metres), this bamboo 'cascades' or droops outwards at the top, meaning it might just block some of the view. It is also one of the few bamboos available which will simply form a large clump, and not spread by runners everywhere, so it needs no root rhizome barrier. Once it's tall enough to do a good screening job, it will block out some of your sunlight in the late afternoon, unfortunately. There's no real advantage to planting it the other side of the door too -it would block the shed window and wouldn't increase your privacy, but planting two either side would look more balanced.