Would it be possible to measure the distance to a black hole of known mass by measuring the radius of the most redshifted radiation that is detectable? The idea is that since the point where that radiation vanishes is related to the gravitational potential difference between the observer and the point of emission of that radiation, we should be able to get an estimate of how far we are by observing how far into the black hole we can see (assuming that for a given mass and an observer at infinity that radius is fixed).
Yes. If you could spatially resolve a black hole and measure the redshift of radiation at a given angular radius, then that would yield the distance to the black hole.
This works because the amount of redshift depends on the ratio of radius to Schwarzschild radius, but the latter is known if the mass is known. Thus the redshift yields the radius. The distance is then given by the radius divided by the apparent angular radius.
There will be complications due to gravitational lensing and any orbital velocity around the black hole, but this does not stop it working in principle.