The white hole is also a hole in space-time and would probably have equal gravity as the black hole. So both holes exert gravity pull on each other. The white hole also has a repelling horizon which pushes. So with just taking these forces into account, a particle placed in the middle might be pushed against the black hole and swallowed.
Such white hole repelling force might seem to be some kind of a new force of physics, and we might not quite know how it works. And if the white hole repel, and if there is symmetry, as it often is in physics, would there be some extra pull besides gravity from the black hole?
Anyway even without these strange forces, energy and matter streaming out of the white hole might also push the particle in the middle against the black hole.
A black hole white hole binary might also resolve the stability issue of wormholes. A wormhole is calculated to be incredibly unstable and would easily collapse, and need a kind of opposite energy to keep the "funnel" open. Exotic matter is one proposal to keep wormholes open. But a binary with oppositely directed wormholes might provide the perfect opposite energy which might stabilize both wormholes, so black and white (worm)holes might then stabilize each other. If wormholes exist in space they might then be such opposite black/white binaries, the white hole would then light up, and the black hole would be far more difficult to observe, as it is a dark wormhole entrance.
I researched the black hole in the center of our galaxy called Sagittarius A*, and could not find a single evidence of it being a black hole, but a dozen of observations pointed at it being a white hole, isn't that odd?
And in the galactic center we could really need a new force, as the orbiting stars does not seem to behave according to Kepler and Newtonian mechanics, as most stars have about the same speed around the galactic center, about 220 km/s, like our Sun. So we postulated dark matter to solve this, but dark matter is both hard to find and hard to distribute correctly in the models. So maybe dark matter is not matter, but instead a strange force with tremendous energy?