I am not sure that is the way to look at it! I get the feeling that you are syncing the dome to the telescope angle when you start for the night.....though I might be very wrong with that assumption.
You need to look at the dome and scope (read mount) as completely separate....it is just the funky geometry that brings the two together for the night.
Say you have your scope pointing low at 179° (almost south). Picture how a GEM mount might be sitting.....your scope will not be pointing out from the center of the obs....but will be off to one side. So the angle reported by the dome will not be 179°....it has to take into account the funky geometry as you call it.
The dome (software) needs to know where the shutter opening is....so the dome needs to be aligned with NSEW before it ever tries to follow the mount. It must have it own reference point - not just be aligned with the scope when starting out.
So my question at this point is.... have you got the dome home or park position properly fed into the dome software? Does the dome "know" what actual angle it is pointing at? (Not just lined up with where the scope is pointing). If you have that OK then the 'funky geometry' will ensure that it points to the right 'offset' from the angle that the scope is pointing to.
The 'sync' button is used if you notice that there has been any slippage between the dome drive and the dome actual angle. E.G. if your dome park/home position id definitely 350° but you notice it is showing say 355° when you park it up....then it is easy to sync it back to 350° (using that button) and things will be correct when you start up again the next night. You would not normally use the sync button to sync the dome to the mount angle....at least that is how I have worked things with my GE Mount and once I got the funky geometry number right...I have never had a problem since.
The hard part in all of this is getting the dome angle right...