Lids stack, so the one you want is always at the bottom. Standing them on edge solves it in one move.
The rack
Two rails with dividers, expandable from 28cm to 46cm. Set the dividers to the lid thickness you actually own.
Which drawer
It needs 12cm of depth and a drawer at least 30cm wide. Measure before ordering; this is the one thing people get wrong.
Materials
Powder-coated steel rails, ABS dividers.