What is wrong with my calculations of Jupiter's apparent magnitude?
I think that should be $+ V_{Sun}$ in your second equation, not $- V_{Sun}$.  Otherwise, a fainter Sun (larger magnitude) would give you a brighter Jupiter (smaller magnitude).   Those need to go in the same direction.
Applying that change (assuming the rest of the calculation is right) would increase your answer by $2 \times 4.83 = 9.66$, which gets you much closer to the right magnitude.
Your estimate will now be a bit on the faint side - not sure why - but at least a lot closer than before.