Datalink frequency shift.
The probe is in communication with a relay station live streaming the event back to NASA and eager nerds like me. The communication is bidirectional, and the probe is constantly acknowledging packets received. (Think TCP protocol but tweaked for space.)
The constant stream back of acknowledgements keeps the communication channel constantly active, allowing the probe to measure the frequency shift of the data link.
Because it knows that the relay is far enough away from the black hole (and traveling slowly enough) that time dilation isn't an issue to N significant digits, it can calculate the exact time dilation from the frequency shift of the known communication frequency.
Eg the 5Ghz radio signal is arriving at 25.453Ghz - so that relay stations clocks are running 5.0906 times faster than mine. That's means my clocks are running at 19.644% of real time. You now know your time dilation factor to as accurately as your communication unit can measure the changing frequency.
Assuming you're not orbiting at relativistic speeds, from this, you can calculate your approximate distance from the black hole.
If you are orbiting at relativistic speeds, ie, you're getting in that close, you can measure the change in time dilation (via frequency shift) at regular intervals (in probe time). I'm not smart enough to be 100% certain this will work without coding a simulation, but my gut instinct is that there will only be one valid orbital ellipse possible for a given sequence of time dilation measurements.
After you consider:
- time dilation from the gravity of the blackhole and the probes distance to it.
- time dilation from the speed of the probe.
- doppler shifting based on the relative velocity of the probe and the relay.
That sequence of changing times should give only one ellipse and your phase on that ellipse. From that, you can do your orbital maneuvers.