Could we verify the structure of a black hole by observing an orbiting object? Can we determine the physical structure of a black hole by observing its gravitational effect on objects in orbit?  There are three possibilities that I see, and would like to test:


Singularity at the center - The traditional view that there is a singularity at the center (e.g. all mass is concentrated in the center).
A structure where the black hole is filled throughout with matter inside the event horizon - My proposal, due to gravitational time dilation slowing / stopping time (e.g. a singularity will not form until an infinite amount of time has passed).  If the answer to my question on this subject here is correct, we should observe either possibility 2 or 3:  Does matter accumulate just outside the event horizon of a black hole?
A hollow shell, with matter only on / near the outside edge of the event horizon - My proposal, combined with the "vapor bubble" behavior, as hinted at in some of the answers to the linked question.


Would these three possibilities differ gravitationally in ways that could be observed (e.g. by observing orbiting stars), or would they be identical from the outside?  If they would produce different results, I would love to see what result it is that we actually observe!  We have observed stars orbiting the super massive black hole at the center of the galaxy.  http://scitechdaily.com/astronomers-discover-star-orbiting-the-black-hole-at-center-of-the-milky-way/
The comments from @userLTK, and @Lacklub are correct. 

Lets assume there is an object of radius $R$ and mass $M$, from a Newtonian point of view, if you are at another radius $r$, such that $r &gt; R$, then there is no difference in the gravitational field  experience by an object at $r$ if the mass spread across a shell of radius $R$ or if its concentrated anywhere between $r=0$ and $r=R$. GR doesn't do much to change this, and in fact if $r &gt;&gt;R$, then the result is of course exactly the same.

Now to black holes, all taken from from Wiki https://en.wikipedia.org/wiki/Black_hole

Physical Properties
"The simplest static black holes have mass but neither electric charge nor angular momentum...This means that there is no observable difference between the gravitational field of such a black hole and that of any other spherical object of the same mass." 

So basically if you're outside the event horizon (not that'd you know where it was) you're experience with the black hole is the same as with a planet or star of that mass.

Again from same wiki article:

Singularity
"At the center of a black hole, as described by general relativity, lies a gravitational singularity, a region where the spacetime curvature becomes infinite... It can also be shown that the singular region contains all the mass of the black hole solution". 

The "it can be shown" references page 204 of Carroll, Sean M. (2004). Spacetime and Geometry. I don't have my copy with me right now so I can't look it up, but I would say I remember reading at in Carroll back in the day. 

Finally, let me just add that, once something passes into the event horizon of a black hole, there's no getting back. So we really have to ask ourselves how any information about the "stuff" inside the event horizon would come to us? I like your idea of getting some indirect evidence and perhaps gravitational waves will shed some light on this topic but I don't suppose there is any known way to getting direct access to anything beyond the event horizon.