I'm new to concertina, so please take it easy on me....
>> When the concertina is just sitting there the bellows will expand a tiny bit.
I don't think this will hurt the concertina. The only thing I'm concerned about the concertina's bellows is that in transit, if the ends of the concertina are not blocked, the bellows keep moving in and out as if the concertina were being played. That may result in pre-mature wearing of the bellows.
Most of the hard-shell cases have the ends of the concertina blocked. So moving the concertina in its case is not a problem.
>> The button accordion when in resting position is on it's side and the weight of it keeps the bellows closed.
I'm not sure what kind of accordion you were referring to, but I play CBA. What recommended to me by the people who know much more than I do is that the accordion, especially one with a tone chamber, should be stored in the "playing position." This is to prevent the pre-mature sagging of the valves.
All of my CBAs, except for the Roland V-Accordions what have no valves, are stored upright in the playing position.