Are all of the characteristics being measured with the exact same hand held gage? then I might agree with Bob's suggestion.
However, since you don't mention what the product is, what the characteristics are or what gage(s) are beign used, my generic answer is that each characteristics/gage combination needs to be qualified individually. My organization produces medical diagnostic instruments and each critical characteristic is qualified separately and completely. While appliation of MSA to such a complex device might be obvious, I also have each critical dimension of pieces and parts seperately qualified. An example is of a plastic pipette tip with 6 critical dimensions. during the R&R phase for thsi tip we discovered that a critical dimension, measured with an optical comparitor - wasn't being measured correctly by the supplier. Unfortuantely their bias was in the direction of a critical failure mode.
Ther is a reason we do measurement systems analyses and it's not just to check the box for your customer report. It really is a matter of pay me now, or pay me (1000X) more later...