You don’t need to cancel out the velocity with respect to the sun though, since you aren’t trying to land on it, or even trying to enter into a circularized hel

You don’t need to cancel out the velocity with respect to the sun though, since you aren’t trying to land on it, or even trying to enter into a circularized heliocentric orbit. The first two burns of a bi-elliptic transfer (as if you were attempting to insert to a nonexistent Earth orbit at the sun’s altitude) should put you on the right trajectory I think?