If we were to locate a pulsar in a close elliptical orbit around a black hole, could this reveal significant information about the black hole (e.g. time dilation / structure)? Could this even tell us the structure of a black hole (e.g. if it is a singularity, or a sphere formed around the event horizon, or something else)?
Yes. And we are actively searching for these systems, e.g., with the Pulsar timing array and "soon" with instruments like SKA.
From the Astro2020 Science White Paper on Fundamental Physics with Radio Millisecond Pulsars:
A broad class of alternative theories invoke mediation of gravity through both tensor and scalar fields, whereas general relativity relies strictly on a tensor-field description. A key prediction from tensor-scalar theories is dipolar gravitational-wave radiation in compact binary systems with large differences in component binding energies (Eardley, 1975).
Future improvements to tensor-scalar tests will also come in the discovery of pulsars tightly orbiting stellar-mass black holes. Shao & Li (2018) recently estimated that a small but detectable pulsar/black-hole binary population – between 3 to 80 such systems – resides within the Galactic. disk
And about super-massive black holes:
A key science goal for future radio-astronomical observatories is the discovery and timing of radio pulsars in orbit around the supermassive black hole residing in the center of the Milky Way galaxy (e.g. Bower et al., 2018). Recent projections have shown that an entirely new class of tests can be achieved with pulsar/black-hole orbits shorter than 1 yr in period, even if only one pulsar is discovered and yields low timing precision (Liu et al., 2012). These tests will directly probe mass and spin properties of the Galactic-centre black hole, as well as the validity of the storied “nohair” theorem (e.g. Will, 2008) with measurements of mass, spin and the quadrupole term of its gravitational potential.
So, you see, these source will be great for testing GR in general and studying the structure of BHs (or whether there is any) in particular.
Btw, such gravitational time delays, are often called Shapiro delay .