I always harvested our juice grapes (concord) soon after full coloring, but I recently met someone who leaves theirs on until they begin to soften, many weeks later.
Is there a certain time to harvest grapes, that gets the best balance of shelf life (before processing), flavor, and sugar levels?
It depends on what you're going to do with the grapes, and what your preferences are. The biggest factor between the two mentioned harvesting techniques is the sugar content, which will become more and more concentrated as the fruit ripens, even after coloring.
The fruit will store longer if picked earlier, and will increase in sweetness off the vine. If you want a sweeter juice, pick later if possible, as maximum sweetness is gained by full ripening on the vine.