Actually, as the most recent examination (a big pdf) of the Pioneer anomaly by the those doing the tracking makes clear, the JPL tracking model does include relativistic effects - see section IV, A.
However, there’s nothing in there to suggest that these effects are actually significant in practice. State of the art in generating an ephemeris will be such that the programmes incorporating the corrections will be available to the JPL anyway and it’s probably easier just to use them.
Well, I did say my calculations were.“extremely rough”.
Uh, so there.