For the Vega measuring exercise I’d suggest a Self-levelling Rotary Laser as this can afford above floor level measurements without physical contact with the exhibit and its ensuing concerns. The red-beam versions are most suited for daylight conditions and the green-beam for interior lighting. Tripod mount the laser and progressively adjust its height to position the laser beam on the desired exhibit spot, then simply steel tape measure the height of the beam away from the exhibit, optionally using the electronic beam-detector which can resolve to around 1mm. Thus there should not be any need to use a surveying staff. To restrict the rotary beam from unwanted areas of the room merely add a baffle, like a hand/pocket tissue, around the rotating head but not touching it.
To measure the off-floor exhibit wheel heights hold a spirit level under the bottom of each tire and measure the height of the top of the spirit level at a point clear of the exhibit. Or if the wheels are close to, but clear of, the floor use external engineering calipers, which come in various sizes.
If the exhibit is non-level it might be prudent to tape measure the height of each wing tip in case someone subsequently attempts a computer CAD re-orientating exercise; a daunting challenge!