An extremely interesting thread...
I'd heard the story about Sherman being offered the throne before, but learning that no one has been able to source it outside of a Flashman short story is intriguing. Fraser loved using those types of oddball historical tidbits to insert an air authenticity to the Flashman Papers, yet he didn't feel the need to footnote the anecdote. It's a nice little mystery, isn't it?
Returning to the OP's suggestion, looking at the last half decade or so of the OTL Sherman's life reveals a man who was fading both physically and mentally. There's a well known story about Robert Louis Stevenson visiting the retired Sherman in NYC.
The visit went well at first as Sherman had enjoyed the author's books and was looking forward to meeting him. However, as Sherman indulged more and more in reminiscing about his campaigns, he began to forget who Stevenson was and had to be reminded several times. Sherman would ask who Stevenson was while referring to the younger man was "one his boys" in a reference to the many veterans who visited him.
This fading Sherman who somehow accepted the Bulgarian throne in 1879/80 would present the new nation with a major political crisis within a few years.