One way of looking at this is not to try to draw a "straight line" between the two points, but instead to consider the arc length of the circle that they both lie on (centred on the black hole). If the two points are close together (i.e. d is tiny compared with the radius), that's a good approximation anyway.
Choose your coordinate system so that both points are on the "equator" [itex]\theta = \pi / 2[/itex]. The circumferential distance between the points is just [itex]r (\phi_2  \phi_1)[/itex] and both particles move along worldlines of constant [itex]\phi[/itex] and [itex]\theta[/itex]. So the circumferential distance in the limit as they approach the event horizon is just [itex]r_s (\phi_2  \phi_1)[/itex] where [itex]r_s[/itex] is the Schwarzschild radius.
Oops, Ich beat me to it!
