One thing I have found interesting, is how the mind/brain handles the transition from one "transposing" horn to another. My brain automatically switches into Bb mode as soon as I feel a tenor and soprano in my hands. Same when picking up alto, sopranino or bari. I hear notes and my fingers (generally) go to the right places - that comes from spending enough time playing a given instrument that hearing/playing notes becomes second nature.
Playing a C-mel or C-soprano kind of blows my mind for a while until I spend some significant time playing them. Just playing a scale causes a feeling of disconnection with the horn - pitches come out that the brain doesn't associate with the fingering. Even worse, there is a disconnect when improvising - hearing a line or melody but the fingers don't naturally head for the places to create those notes. I guess for some folks the jump to a different transposing instrument would be an easier process than for others. It is possible than some players don't have such a strong association between fingering and pitch and could more easily play without the distraction of the pitches being at different fingerings then the brain has programmed.
This is all to say, that I expect the same disconnected would occur when you first start playing the G-mezzo. Maybe it might take only a day or two to incorporate its unique pitch/fingering transposition into your playing process.
Another interesting thing about an instrument in G is that you will end up playing in more "flat" keys than "sharp" keys. A song that is in the concert key of "G" would be played in E on an alto but in C on a G-mezzo. A tune in Eb would be in F on tenor/soprano but in Ab on G-mezzo. Its a game changer when playing in "guitar keys".
Anyway, please keep us updated on your adventure with the G-mezzo!