I'd like to keep it neat without anything sticking out and being visible from the cupboard. None of these devices had any issue with my previous computer that was placed in the exact same manner, so I imagine it must be possible to get it working here. But could it be that the network card is simply trash and can't output any stronger? In that case the same kind of USB BT adapter could simply be placed in the cupboard. I think that would be an acceptable backup solution if I can't resolve it with existing hardware.
EDIT: I should add, keeping the door open does not improve anything. The cupboard is wooden.