It could certainly be stable; it's possible for a star to be ejected from a galaxy and still have other objects be gravitationally bound to it. Observations of the hypervelocity star HE 0437-5439 indicate that it was likely originally a triple system. During an interaction with Sagittarius A*, two of the stars were flung away at a high speed bound to each other as a binary system. While they eventually merged after tens of millions of years, this was due to normal stellar evolution -- one swelled up into a red giant and engulfed the other.
A similar situation could apply here. Imagine you have an s-type binary system, where the planet is tightly bound to one star (so there are two stars and a planet, as opposed to three stars). During an encounter with Sgr A*, the planet and the star it's bound to are propelled to high speeds, eventually leaving the galaxy. Since hypervelocity stars can reach speeds of ~1000 km/s and the Galactic disk is ~100 kpc wide, it could leave the disk within ~100 million years -- less if the system is ejected at an angle out of the Galactic plane.
Of course, an encounter of this sort could very, very easily lead to disruption of the planet's orbit during the close pass to the black hole -- four-body interactions are, uh, not pretty -- but there's a chance that things could turn out the way you want. The bottom line is that invoking this scenario as a background to explain your rogue system is reasonable; the outcome you want is definitely a plausible one.