Best advice so far, subject to the extraneous-conductive-part-TNC-S-export issue.Or rather than wasting money and time on all that lot, just use 2.5mm SWA from a 20A MCB in the house CU, and connect to a 2 way CU with RCD in the shed.
If you can only connect via an RCD in the house, don't bother with the CU in the shed at all - use metalclad sockets and connect the SWA directly into the first one. Connect the lights via a 5A switched FCU.
But if you're going to bury the cable then unless digging trenches and repairing the garden is a hobby of yours I'd advise installing 6 or 10mm² SWA so that when you decide to install a hot-tub you won't have to re-excavate.
If you have room next to the CU for an adaptable metal box 100mm-125mm size you may find it easier to terminate the SWA in that and just take the cores through to the CU rather than trying to gland the cable directly into the CU.
Are you in England or Wales, planemad?