Which climber to choose?

Oh dear, Sandie - that isn't very deep for a 'perennial' climber. Could I suggest that you grow annual climbers such as Ipomoea or Dicentra scandens. Alternatively, there's a not-very-vigorous Clematis which might cope - C.florida var.sieboldiana. This is white with lots of purple stamens. Campsis x tagliabuana (Trumpet Vine) is vigorous and would cover the wall, but I am not sure whether the depth of the box would suit its root ball. Good luck, anyway.

I don't have any recommendations, just a word of caution. My neighbour just spend a bundle of money on new gutters, and roof parts because a climber "kept on climbing" and destroyed part of the house.
If you use a climber, be sure to keep it under control.