Adjective "butch" (deeply or stereotypically masculine-looking and -acting, applied to both men and women) made into a jocular noun. "Butch" is nowadays almost always an adjective ("I love your butch new leather jacket"), now that the old use as a noun to mean "boy vendor" (newsbutch or news butcher) peaked during the 1950s and is now archaic if not completely obsolete. The closest nominative uses of "butch" are adjuncts or idioms like "butch haircut" or "Harold is a butch top, not a femme bottom."
"Masculinity" means possesion of masculine traits or appearance, while "butchiosity" introduces a note of irony: the appearance or trying to act in a conventionally or socially-acceptable masculine manner expected by the straight world.
"Butchness" is problematic, as it is not consisently used to mean the quality of visible masculinity; instead, the user is often forced into grammatical adaptations like "He behaves in a butch way," or "Sometimes he overcompensates and out-butches himself."
"Butchiosity" is almost certainly a back-formation inspired by the 1977 Woody Allen movie ANNIE HALL (screen play: W. Allen and Marshall Brickman): "Was it heavy a rock concert? Did it achieve . . . heaviosity?"
"Your buddy ought to butch up his act, or he'll never make it in the suburbs." "I don't think it's so much a matter of butchiosity as the fact that he hasn't been around suburbanites since going to college and knows next to nothing about kids, lawnmowers and pro sports."
"Who is that great-looking hunk?" "Oh, him satirically, "her". He looks like a tower of butchiosity but on the inside there's a major flaming queen screaming to be let out."
Compared to "masculinity," "butchiosity" introduces a note of irony: the appearance of masculinity, or trying to appear conventionally masculine. Machismo is lexically coherent as a loan-word from Spanish but has connotations of an ingrained, often unacceptable and patriarchal behavior, overmasculine from the start; or old-fashioned notions of being a man. Instead, "butchiosity"comes with a twist: it can imply a trying-too-hard, self-conscious or overstudied presentation of an image of masculinity, especially in an attempt to meet the straight world's expectations of same.
An act not a trait, as it were.