When it comes to the orbital mechanics of this scenario, one thing to note is that when it comes to oberth effect and raising the orbit around the Sun, it's the total velocity relative to the Sun that matters, i.e. the Earth orbits the Sun at 30km/s, and the probe orbits the Earth at 8km/s, giving 38km/s to work with, the bare minimum ejection burn to intercept Jupiter then adds 6.5km/s and the probe is up to 44.5km/s, now the question is are you better off burning on top of that 44.5km/s, or burning at Jupiter?
Jupiter orbits the Sun at 13km/s and the probe should swing by at about 60km/s if it gets as close to Jupiter as possible, potentially a combined velocity of 73km/s. The tricky part is that at closest approach to Jupiter the probe probably won't be travelling in exactly the right direction for a prograde burn to enjoy maximum oberth effect with respect to the Sun - at Earth precise timing can arrange this, but when you're swinging by one planet and need to actually encounter a third planet there are constraints on the first encounter which will result in the oberth effect not being maximized, unless you wait a long time for the planets to line up in exactly the right way and when it comes to long period inclined and eccentric orbits like that of Pluto, that's going to be a long wait.
So this is not enough to rule out the possibility that it could be more effective to do some of the burn in Jupiter's gravity well, it's just to say you get an awful lot of oberth effect already by virtue of doing the burn deeper in the Sun's gravity well. My hunch is that if all you care about is maximizing heliocentric velocity you'd be slightly better off doing the burn at Jupiter, but if you wish to actually encounter a third planet you'd tend to be equally well served by completing the burn at Earth.
And there are also the practical constraints: completing the burn at Earth uses only the one stage for the entire ejection burn. A Jupiter burn would require an additional solid stage or a reliably restartable engine and a no leaks guarantee. And if the burn is in any way messed up you've just lost your best option for cheap course corrections, which is fine-tuning the encounter with Jupiter.