You can access the BA lounge while staying airside, however it is a very long walk.

The Admirals Club has two shower rooms which are perfectly nice, and will be serving a light breakfast (typically eggs, oatmeal, cereal, bread products). I can’t really think of any place in there to leave your bag for the day, however.

Updating to add a picture of the shower room; there are at least two like this, maybe three. I’ve never had to wait for one in the morning.