Odd that it's so difficult to find an answer. MLK Jr's father switched to Democrat, after the following instance, but MLK Jr "made no endorsement". The event below was 8 years before his death.
The idea that MLK would be a Republican at some point in his life is at least credible. The problem is that both parties have changed significantly since that time, so the point isn't particularly relevant beyond being an interesting trivia question.
Some of the most right wing factions of the Republican Party come from ex-Democrats that bolted because of national defense issues and civil rights issues.
Some Republicans wouldn't be very upset if the Democrats would take them back.
Location of credible evidence?
I don't know of any source that King was a philosophical "Republican," especially when he started developing his ideas about Civil Rights and imperialism.
By that point it is clear that he philosophically differed from republicans.
There is nothing in the article above stating he was a Republican; I think Republicans would have made a big case out of it if he was.
Alabama had a Democratic governor from 1874 to 1987. Patterson (59-63), Wallaces, both George (63-67, 71-79) and Lurleen(67-68) were defenders of segregation while Brewer (68-71) not only cooperated with segregation plans, but actively courted black voters in his losing campaign against Wallace (Brewer succeeded Lurleen Wallace when she died - he never won a gubernatorial election).
Georgia had a Democratic governor from 1872 to 2003. Vandiver (59-63) was a defender of segregation. Sanders (63-67) cooperated with Kennedy/Johnson plans for desegregation.
Mississippi had a Democratic governor from 1876 to 1992. Coleman (56-60) cooperated with segregation. That also caused him to lose elections in 60 and 64 to segregationists Barnett (60-64) and Johnson (64-68).
You could go through several Southern states with the same pattern.
If you have a cause like civil rights and that cause trumps other considerations, it wouldn't be outrageous to think more help would come from Republicans than Democrats. Of course, that would be tempered by the fact that Republicans couldn't win elections in the South back then, so they weren't in a position to offer much help on a state level. On the other hand, if you can't win elections and a large voting block is available .....
All I'm saying is that not openly declaring himself Republican or Democrat held some advantages and which party became associated with civil rights in the South was an open question back then.