You asked an interesting question:

If (arguendo) anti-matter anti-gravitates, then

> Does it not then make sense that in-falling anti-matter would have an evaporating effect

Yes, kinda. More precisely, it would depend on how anti-gravitation works exactly, i.e., how anti-matter couples to gravity differently from matter. So even more precisely, "we don't know but can model various remotely plausible ideas with current theory" [a concrete attempt to model below at [1]].

One approach to model anti-matter anti-gravitation is to have a second metric which anti-matter feels but matter doesn't. Bimetric theories have been studied for several decades, and the problem is that one has to hide the effects of the second metric or we get a very different night sky than we observe. So while bimetric theories aren't ruled out, the effect today (as opposed to the very early universe) is extremely small. A tiny present-day effect would be interesting in that it would suggest a possible mechanism for the overwhelming dominance of matter in our part of the universe: a decaying secondary metric could strongly segregate matter from antimatter in the very early universe (preferentially collapsing matter together with other matter, and antimatter together with other antimatter), while not interfering with today's observations of high-redshift galaxies or laboratory experiments to date.

In a bimetric setup, a black hole would source something akin to Schwarzschild for the dominant metric, and something akin to anti-Schwarzschild (for matter) as a secondary metric. We could contrive a secondary metric that causes antimatter produced in the hot accretion disc to receive a dynamical boost that transfers extra energy-momentum into the antimatter and flings it off towards infinity. In that case, anti-matter that anti-gravitates would have an evaporative effect on the black hole.

There is of course no observational evidence favouring a strong secondary metric anywhere in the known universe, and it would be an awesome surprise if forthcoming black hole telescopy found any. Meanwhile people are working on ruling out a weak secondary metric in laboratory settings.

There are other approaches to theoretical modelling of anti-gravitating anti-matter, but they generally need a dynamical suppression of the effect at modern times and large mass scales, for the same reasons. Such approaches typically look really highly contrived on their faces, and of course suffer from lack of evidence, since we don't have a good view (with current technology) of the universe before the cosmic microwave background formed.

We can be pretty sure that anti-matter doesn't anti-gravitate at all from nuclear physics (electron anti-neutrinos in beta decay, and the antiquark halves of mesons) in astrophysical observations (you get a lot of both in supernovae (SN), and anti-gravitation of anti-quarks or anti-neutrinos would lead to a noticeably differently massed SN remnant). Of course we would like to be very sure with many more lines of evidence. :D

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[1] Modelling anti-matter anti-gravitation is not a popular academic area of enquiry, mostly because there's no evidence for it, and there's already lots of model-building in high-energy physics (and string-theoretical gravitation) where one has to "hide" side-effects that aren't seen around here-and-now somewhere in the universe (distant past or future, well outside the Hubble volume, or in extra dimensions) that solve a broader range of problems than introducing anti-matter anti-gravitation.

However, on that front, see Hossenfelder's blog entry http://backreaction.blogspot.com/2017/04/why-doesnt-anti-mat... which refers to her paper https://arxiv.org/abs/gr-qc/0508013 which does not deal with anti-matter as you conceive it in your question, because she proposes duplicating the ENTIRE Standard Model (including its anti-neutrinos, positrons, anti-quarks, etc.) that feels gravity normally with an exactly similar set of quantum fields that is feels gravity as a repulsive force. She then introduces an anti-gravitation which this second set of standard-model-like-fields feels as attractive, while the Standard Model is adapted so that its constituents feels it as repulsive. She likes toying with mathematics, and this is a classic example of building a model and chasing down its consequences just for fun -- she certainly does not propose that this is a plausible model of nature, since it breaks the equivalence principle in a way that conflicts with tests of it.

In her model for technical reasons the anti-gravitating extra set of particles have to be suppressed from Hawking radiation, and therefore are not evaporative of black holes. (They would be repelled from black holes if they were brought from outside a BH towards a BH, though, just as they would be repelled from the Earth if brought near the Earth). She does not elucidate a mechanism for this suppression.

Finally, Hossenfelder's model is single-metric, rather than the sketch of a bimetric approach that I described above. However, it's also closer to the exact wording -- if not the spirit -- of your comment.