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?
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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.