DIY'er rather than pro ?
Will the 20a main board fuse run off an rcd rail?
Ideally there should be an rcbo / rcd circuit control for the added safety reasoning and to at least TRY to comply with 17th ed.
Why not run the 2.5mm to a fused spur rcd unit rated at 13a and then daisy chain on to a 2nd fused spur rcd unit rated at 7a. You then get some safety, and isolation of the sub circuits (light and sockets) in the garage.
The pro method would be a feed direct from the main intake via a henley block and then provision of a sub board (builders CU) in the garage with rcd + breaks or a couple of rcbos. However, if you don't have any intention to test or can't get access to a mftu doing it the correct way and not testing is worse than using the fused spur rcd units I've suggested.
Flame suit is on, awaiting i/c BAS missiles.
I'd also add that such external works shouldn't be done DIY since Part P says so !
Finally conduit along a fence is iffy since you will have problems securing the saddles, the plastic is on the edge of being acceptable and a fence isn't considered a permanent structure- as in the posts rot, and or wind could pull the whole lot down.