But in the view of the bugbear on the receiving end, its toughness hasn't changed a whit and nor has anything else about it: it's the same bugbear. All the changes that make the combat play out differently have happened on the attacker's side. All of them.
All the changes that make the combat play out differently are game-mechanics abstraction, not the fiction. And, as we'll see in a moment, the combat probably doesn't play out differently in any meaningful sense.
And to reflect this in the fiction, nothing about the bugbear's numbers should change at all.
The numbers are not the fiction. The fiction is what you get only after all the numbers are crunched, and the thing is summarized with the numbers removed. No game stat appears in the fiction.
Plus... the bugbear in the book is a guideline. I can represent bugbears in my game any way I want - the Monster Manual numbers are merely one mechanical representation. I can have bugbears that are smarter than average, or weaker, or smellier. I can put up a bugbear warlord that has the stats of a 20th level fighter. I can make a sickly bugbear that has the stats of a goblin. Or, I can make a mook bugbear that has a high AC, and one hit point. We are not beholden to the MM.
There is a general expectation about how bugbears are not usually pushovers for low-level parties, yes, but so long as the experience roughly matches that most of the time, we are okay. The only worry we have is if the first attempted hit downs him outright - but for a higher-level party, narrating that as an excellently aimed shot to a vital part is still generally acceptable. If we have a crowd of them, that happening once or twice is okay. It is only if I have a horde of them, and they are taken down like chaff, that am I violating the expected fiction.
It isn't like we are talking about a well-known individual with an established backstory who might be asked to match those storied deeds in play. Correct me if I am wrong, but we are probably talking about a generic guard a higher-level party will encounter once, and probably not have a conversation with other than, "Die, hairball!" If I stat it out as a regular bugbear, the PCs are sure to get past it after a round or three. If I stat it out as high-AC and 1 HP, the PCs are sure to get past it in a round or three. The fiction is the same, either way - some nameless bugbear delays them for a round or three, and dies an ignoble death bleeding on the floor.