Would a spinnet need tuning every time it was moved?

Very likely - and it would cost a pretty packet too, even in kit form. Indian harmoniums are cheap and effective, though essentially monophonic because you use the left hand to work the bellows, as with a medieval portative. Portable pedal harmoniums are quite hard to come by, especially the ones that are truly portable! You can get an extra small Indian harmonium, like a melodica with a bellows - but I'm not sure how much better these would be than those toy 8 button accordions you get in most music shops these days... I have one & love it dearly, but I have a particular fondness for musical toys.