Hm... I'd bet that you have a better chance to find high quality freshwater pearls died black rather than high quality akoya died black. It makes sense to me that any akoya good enough to sell bleached white, will be left white. Possibly the same applies for freshwater pearls, but with their lower cost across the board, exceptions are possible. At least that's what I came to believe with the scarce info around... Run a search on this forum for 'black freshadama' for an example. There must be others as well.
Hope my 2c helps
PS: Since you are interested in tinted pearls,
THIS might be fun (a video showing one way treatment is done).
THIS thread explains where it is comming from...