I know that there is plenty of theory to predict the size of a neutron star: my question is whether or not there are any reliable size estimates based on observation. Please note that, to be reliable, such an observation would need to have also a good estimate for the neutron star's distance from us, which must necessarily be reckoned from some independent source, i.e., it must not be reckoned from the apparent size of the object, or from theory, otherwise one is guilty of circular reasoning!
Yes, the neutron star RX J1856.5-3754 has an observed* radius of 17 km. After accounting for general relativity, its actual radius is calculated as 14 km. It is not the only neutron star whose radius is known.
* It's worth mentioning how astronomers arrived at this radius. Technically, fitting the X-ray data to a blackbody arrives at a calculation of a 5 km radius, which seems unusually small, so for a time it was believed that this star may be something more exotic, like a quark star. However, after atmospheric modeling based on temperature, a more accurate estimate of the emission radius was derived to be 17 km. Once gravitational redshift was accounted for, the value of 14 km was yielded. See Magnetic Hydrogen Atmosphere Models and the Neutron Star RX J1856.5−3754 - Wynn C. G. Ho, et al.