If Doomslayer/Doomguy got into Smash, whatever Sakurai decides to call him will probably be what he is know as by the fanbase in the future.
I posted something like this earlier, but I don't think people understood why so Im making it more clear here:
Well, if he was called Doomslayer in Smash, it would feelstrange to call him Doomguy in discussions about Smash, wouldn't it? It's like saying "I main Ren Amamiya in Smash" or "Red's the best character in Smash!" Sure its not wrong to call the characters that, but it just feels... off.
Obviously, Smash is a bigger franchise than Doom, and there would be a lot of discussion between those two fanbases.
So, if Doom fans, who probably don't really care about calling him Doomguy or Doomslayer see litteraly everyone calling him "Doomslayer," yeah they would feel more compelled to call him by that name