How to carry a bike box from the start to end of the trip?
Some possibilities:


Some searching suggests there's at least one bike shop in Leh. You
could call ahead and get them to save a suitable bike box for you.
You could scrounge some suitably big pieces of cardboard once there
and either buy or bring some packing tape. Properly reinforced, boxes
"made" this way are totally fine for shipping bikes in.
Finally, many airports have bike boxes for sale. The disadvantage
with them in my experience is they're usually the enormous type that
are intended to allow minimal disassembly but are very expensive to
actually fly with.