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).
How elliptic arc can be represented by cubic Bezier 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.