The fic apparently described the 10th Division moving the oin (“That would be the Ōin, a key-type thing from the non-canon DiamondDust Rebellion movie that controls the universe, and a very valuable item that the Gotei 13 gave everything to protect,” explained Lapis).
Wouldn’t it be better if you had a period after "the oin" and no parentheses enclosing Lapis’s explanation?

Then two more mini-Menos Grandes, masamoyto and Hitsogaya, showed up — and we're promptly portaled to the Adoption Agency.
Should be "were".

Take it from an Naruto fan!
Should be "a".

She grabbed the RA from Whitney and opened it to the Mini Adoption Agency.
She opened the Remote Activator? (Should be "a portal".)

Bayonetta stared after the portal as it closed, her guns lowering.
Not being a native speaker, I’m not sure whether this preposition can and should be used here.

HG

YOu forgot to edit the Wikia's Mini page with the Undertale and Bayonetta minis.

1) Sure enough, no sooner had the portal opened up when the screaming began.
Sure enough, no sooner had the portal opened up than the screaming began.

2) I will personally partake in the consequences that may befall you two because of it.
As written, you are saying that Whitney is going to be punished alongside Backslash and Lapis. Unless I misunderstand the context, what you really want to say isthat Whitney will be doing the punishing. As such, you're missing a few words:
I will personally partake in doling out [or some other verb phrase] the consequences that will befall you two because of it.

3) is Enzo, whomever he is, supposed to be hated by the fandom of this Bayonetta game?
Misuse of "whom". The pronoun is the subject of its clause, hence the correct word is "whoever"

4) as well as the guy who made those guns of hers’
Delete the apostrophe.

5) Pou eínai éna kaló agóri?
Error in the Greek. Pou means "where". As shown when I provided the translations for you, the correct word is poios.

6) They were originally Hollows who they cracked their masks
Extra word.

7) “I still feel like something something alcoholic,”
Extra word.

8) The Bleach manga available in most bookstores, and I’m sure Let’s Plays of the first Bayonetta game exist on the Internet
Missing a word: "The Bleach manga is available…"


Aside from that, a few questions:
1) Scabbard Fair the mini
I thought you established that the mini was not Scabbard Fair but instead one of the misspellings of Vigrid. I quote:
“Hold up, Drunky,” said Backslash. “Are you sure that mini-Fortitudo is Scabbard Fair? I think I remember something along the lines of that spelling in the first Bayonetta game.”
He picked up the mini and examined it carefully. Sure enough, its tail looked oddly stiff and stuck up at a weird angle. When he looked back at Cupid, yet another mini-Fortitudo, this one with a bandit’s mask pattern over its human chest-face, was perched on his head.
The two male agents looked at each other, and then both of them facepalmed (or face-winged, in Cupid’s case).
“Oh, Vigrid!” cried Cupid. “The city where the first game takes place! How could we have missed that?!”

2) Lapis's electrokinesis
Multiple times in the mission, you have Lapis sparkling with electricity. But IIRC, her powers were water based, and according to her page on the Wikia, she lost those. Is there something I'm missing?

3) Backslash the elitist
You mention in the A/N that "I feel that Backslash is becoming an obnoxious elitist". How so? Unless there's something I've missed (which is possible); I haven't seen it.

4) Whitney punched her fist,
What were you trying to say here?

I have kind of mixed feelings about the grudge between Lapis and Backslash, though I know it doesn't really originate in this story. On the one hand, conflict is good; not all characters will get along, and major issues shouldn't just disappear between stories. On the other, I feel like the amount of tension kind of distracts from the mission itself. There were times when the badfic seemed to have paused itself so the characters could stand around and yell at each other. It also feels a bit... repetitive? No offense, but a lot of your agents seem to have ongoing grudges with one another that make it almost impossible for them to work together.

Another thing I found distracting was the break in the story to explain about Lapis swearing in Arabic and why she does so. It doesn't really feel necessary to include, or even for Backslash to comment on in the first place. It kind of seemed like an excuse to talk about Lapis's history, even though I don't know that it was relevant to the situation. That said, I think the amount of foreign language swearing got a bit gratuitous at times; suddenly hitting a word in a different language tends to throw the reader out of the story. I'm also not sure about Whitney calling Lapis a "s***head" - she seems more like the "calm rage" type to me than the type to swear, especially given Lapis's age. Though my impression of her may just be mistaken.

I do think you did a good job explaining the canons, as someone who was unfamiliar with them. The Bayonetta stuff was fine - I was even able to get a sense of her personality - and the Bleach terminology wasn't too confusing, which was quite a feat given how long the series is. I would have liked a little more explanation/focus on what the actual plot of the badfic was, but I'm not sure that was actually possible.