I would like to calculate the length of the semi-major axis of Juno's initial heliocentric ellipse from launch to aphelion based on the time from launch to aphelion. Do I use the full 394 days from launch to aphelion or do I subtract the time from launch to leaving Earth's sphere of influence from that 394?
If the perihelion is at Earth, which is what it appears you are assuming since you think half the orbit starts at or around launch, then you don't need the time. You can just use the aphelion distance from the Sun and Earth's distance from the Sun to compute the semi-major axis.
However the perihelion of the orbit may not have been at Earth, if it was injected at some inward or outward angle to Earth's orbit velocity vector, in which case your time is not half of the orbit.
In any case, you should use the JPL Horizons system, in which you can select the Juno spacecraft (id -61) around the Solar System barycenter, and get the orbital elements directly. The initial semi-major axis was about 1.62 AU. That gives a 753-day orbit, half of which is 376.5 days. So 394 days is not half of the orbit.