Alas, you can't exactly represent the orbit of a satellite around Earth with a Bezier curve. You can approximate it pretty closely, though -- "a four-piece cubic Bézier curve can approximate a circle, with a maximum radial error of less than one part in a thousand". (A Bezier curve can approximate an elliptic or hyperbolic orbit with about the same accuracy). See How elliptic arc can be represented by cubic Bézier curve? .
Perhaps you could use current position, the current velocity, and the current acceleration to approximate a parabolic path (pretty accurate for "shorter" times and distances, increasingly inaccurate for "longer" distances). The acceleration is proportional to the sum of all the forces on the ship -- gravitational force, thrust due to rockets, and any other forces. There's a way to convert the starting position, starting velocity, and starting acceleration (which define a parabolic path) to a cubic Bezier curve ... but there's probably some other not-perfectly-parabolic approach that better takes advantage of the flexibility of the cubic Bezier curve. The Derivatives of a Bézier Curve are:
This is for endpoint P0 at t=0 seconds, and endpoint P3 at t=1 second. You'll probably want a single Bezier curve to cover minutes or hours (the Bezier "t=1" location corresponding to the location at, say, 2 hours), so you need to scale the acceleration and velocity correspondingly.
Where to place P3 in order to maximize accuracy? Perhaps you could place P3 at some random location -- say, the same place as P2 -- and then, instead of drawing the full Bezier curve from t=0 to t=1 (i.e., from P0 to P3), you could draw just the early, more-accurate part of the Bezier -- where the location of P3 has little effect -- perhaps t=0 to t=1/8 -- and then re-calculate a new acceleration and a completely new Bezier curve starting from that point. I suspect there may be a better approach.