problem in airing cupboard

Cut 5mm notches into the battens for each cross piece so they will resist slipping on the battens but can be lifted out of the way then you need to remove them.

Alternatively little stubs of dowels on the top surface of the batten to fit into drilled holes on the cross pieces.

In message , Menae writes

Screw the cross-members to pieces of wood slightly shorter than the battens, forming a grid, so that these new end pieces fit on top of the battens. Drill a couple of holes in each of the battens and fit corresponding dowels to the end pieces (or put grid in place en drill right through into the battens, if there's room). Drop shelf grid into place. Remove by giving each end a sharp upward blow.

Alternatively ignore the end pieces and just put dowels on each cross-member to fit into corresponding holes in the battens. More work but less loss of height.

================================ Maybe a bit expensive for your purpose but available here:

Double sided sticky fixers on the ends

Simply screw 2 or 3 battons mid-span, so that the whole shelf is a removable self-contained gridwork. It then sits on top of the support battons you have at each side.

Or have I mis-understood?