I don't think that works as advertised: Any measurement of the current will likely produce a magnetic field itself -- for example from the current in a Hall probe. This current will induct a small, opposite current into the superconducting coil. So the current will go down.

And even if not, what you would need is to measure the change of the field over time. This has finite resolution, so you can't distinguish no resistance from very very small resistance.

It will heat up, as the contact points & probes will have resistance, not sure if there are good ways to correct for that.