would it be feasible to run it under the stairs with a closed door to
the cupboard, or would I just be storing up grief for myself with
condensation and damp building up in the middle of my house?

Yes, although my old Hoover would have been a problem. The Bosch would
handle this. However, you will need some ventilation, as it will produce
some moisture which will cause problem if there is no way at all to get rid
of it. With the adjacent shower room, it would be very easy to ventilate
through there, as shower rooms need ventilation anyway.

I don't know what you've got planned/already existing in there, but just a
simple grille between the rooms might work if the shower extractor is good.
Alternatively, a humidistat fan to vent from the cupboard into the shower,
or best of all, a ducted system with inline centrifugal fan that can suck
from both rooms.

so it shouldn't be difficult to run plumbing and drainage through the
wall between the two.

Don't forget to choose a model with a built in pump that can drain away.
Emptying the tray of water every time is a drag.

As a simple rule of thumb, machines with the water tray at the top have the
pump and can be mains drained, although sometimes they wish to use the
opportunity to hawk an overpriced kit of parts to do so. My Bosch 6920 came
with the "kit" free, and it consisted simply of a length of hose.

Christian.