The Black Sea does have chemocline. It does not mean water does not mix at all, it just means this happens much slower due to high level of stratification and lack of currents. This is the reason USSR considered dumping nuclear waste in Black Sea in the 60s, but opted out and used Kara sea instead, they thought it would be safely buried on the bottom, but upon investigation it quickly turned out it was not the case at all. The pycnocline is also not a sharp boundary where descending from oxygen-rich H2S-free water you suddenly enter H2S-saturated water, there is a gradient. In fact, after the pycnocline, the H2S concentration starts to grow slowly as you descend down until about 400-500m depth after which the concentration drasticly increases with depth and by 1km it reaches the saturation level that remain almost the same down to the bottom.

I don't think comparisons to the Tethys ocean are a good idea, since both water bodies were different and so is the reason for having H2S in them. Tethys at the time was an open sea with much less stratification evident (it became closed much later - in cenosoic I think?) It was also a warm tropical sea at the time unlike the Black Sea with much more biomass in it. The reason for the H2S gasouts was not an influx of organic matter from the continental rivers, it was sudden warming and sea acidification. I am not a geologist and I don't know how and why those occurred, but from what I know, they happened really quick (on a geological scale), immediately followed by massive loss of marine life due to water getting warmer and more acidic and that in turn lead to drastic rises of H2S as warmer water dissolves less oxygen, the bacteria that oxidize H2S require slightly alkaline environment and solubility of H2S itself is strongly dependent on the pH too, eventually becoming a runaway process. None of those conditions are currently available, at least not nearly close to the extent they were at that times (there are proofs that in those periods, surface seawater was highly acidic, surface water pH was less than 6, it is currently about 8 for example). While we live in a global warming period and seas are slowly getting more acidic and marine life is struggling, we don't (yet) witness the same conditions nowadays. It is likely that we may witness them in the not-so-distant future, but Belgrade dumping shit in Danube has very little to do with it.