Is Miss Brill a believable character?

I would suggest that she is. But it's important to see Miss Brill as she sees herself: as a character in a play. Miss Brill is enjoying a Sunday afternoon in the park. Everything seems perfect; the band is playing beautifully and everyone seems to be in their rightful place, acting out their roles as if part of a gigantic stage production. Miss Brill, too, is playing her part in the enveloping human drama, expressing her eccentric individuality through the unseasonal wearing of fur.
Sadly for Miss Brill, her perfect day at the park, her own private performance, is ruined when the two lead "actors" unexpectedly break character. Miss Brill's subsequent retreat to a grim life of loneliness and isolation is sadly all-too-common, especially among people of a certain age. This is what makes Miss Brill such a believable character. Most people will doubtless know someone just like her.
Miss Brill's eccentricities help to make sense of an often senseless world. This world can be a harsh place sometimes, and so it's often necessary for people to indulge in little fantasies to enable them to cope with life's slings and arrows. Some people may find Miss Brill a somewhat absurd, pathetic character. But she, like her fantasies, is completely harmless and unthreatening. And before we judge her too harshly, we should stop to think of how often we ourselves engage in role play, such as when we go online, for example.