Hi Matt. Of the top of my head I would say one easy option would be to do a mix of all the instruments and render it as a stereo pair. Reimport into a new project, use a plugin to raise the pitch and re-record the vocals in the new key.
Alternatively you could send all the instruments to a bus and apply the pitch plug-in to that.
I am guessing the choice would be down to how heavy a load the plug-in places on your CPU - I think pitch shifting can do that.
The "wobbliness" may be down to just that. Is it still wobbly if you render the track?
The other issue is the quality of the plug-in. In my experience no plug-in is totally convincing with large pitch shifts (say more than 6 semi-tones?) but some are better than others with Melodyne often cited as one of the best. I think you can get a demo of it to try but for small shifts (2 semi-tones?) and in a mix, Reapitch may be fine. This also looks promsiing maybe :
https://www.youtube.com/watch?v=_al9aHLkoJU