Interesting problem op.
If I understand correctly, you are outputing both options to pins. Consider the fact that there are different "pin delays" between each output on the silicon die and the corresponding ball/pin.
I assume you are using some sort of constraints to make sure the placement is in the location of your choice?
My approach would be somewhere in the ballpark of implementing two counters, and use them to output a digital reading which would correspond to the delay, by counting a certain number of clock cycles of a inverter with a delay and without a delay. This removes the jitter of the ring oscillators from the picture - because it is averaged over time.
This however adds other problems - as adding delay changes the behavior of the ring oscillator, so another approach is to XOR the delayed oscillator with an non-delayed oscillator, and count a reference clock.
Again though, these approaches give you the affect of routing along with capacitance which changes the oscillator frequency.
All this being said, a ring oscillator may not be the best of choices for this, as it is very sensitive to additional capacitance, and to process and voltage.