PART I: OK here is my reasoning why "Superman: Brainiac Attacks" can actually fit into the DCAU with little to no issues. This post is in no way a response to the WatchTower Database video (which I have not watched since it was uploaded). This will be an image heavy post.
Before we get started, let's get something out of the way:
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For the die-hard fans of S:TAS, presumably the bulk of your readership, let me first say that the movie is not intended to be in continuity with the Timm-verse, despite utilizing those character and background designs. Similarly, the depiction of Lex Luthor was intentionally modeled after Gene Hackman's performance in the Donner films, to dovetail with Luthor's depiction in the upcoming Bryan Singer film. If a die-hard fan goes into 'Brainiac Attacks' with this in mind, it should be a smoother ride -- and hopefully an entertaining one." - Duane Capizzi, writer for the film
It is my opinion that this statement was absolutely intended to be damage control because he wasn't confident in the quality of the feature compared to the series it's apparently "based on". While I'm sure there is some truth to it, I normally don't take official statements like this too seriously. Instead, I look at the actual content itself (which is all fictional anyway) and judge it for myself.
These productions are all very intentional. There was absolutely a reason why in 2006, six years after the "The New Superman Adventures" ended, the team chose to go with these decade old designs. My opinion is either nostalgia (like "Batman & Harley Quinn") OR budgetary reasons. I imagine half-way through the feature, the team lost confidence so decided last minute to make "not canon" with the DCAU. There is a reason why the designs from "Justice League: Unlimited", "The Batman" or "Legion Of Superheroes" were NOT chosen, or even an all new design aesthetic.
For some, that statement alone is enough to satisfy the question. For those who aren't quite convinced, allow me to lay out my reasoning.
First understand that for the DCAU I have worked over a decade on laying out a chronological viewing order based on my own findings. When doing so, I only counted "on-screen media" - minus video games. TV shows, films, shorts, etc. are all counted in mine. Books, comics, novels etc. are not. My timeline will differ from many other's out there, but I am confident that it's as accurate as possible - however, I'm always open to change if new information trumps my current document.
Now, lets get into the meat of this breakdown.
Brainiac was last seen during the events of "Knight Time" where it was launched into space and seemingly destroyed.
In this film, Brainiac arrives to Earth via a Meteor.
A few years later in "Justice League" we find out Brainiac has a base of operations in deep space where he sends out duplicates of itself.
It is my opinion that the Brainiac featured in this film is NOT the original, but one of these duplicates.
While it appears that Brainiac is unfamiliar with Lex Luthor, despite the two having previously met in STAS, it's most likely due to being a highly damaged fragment whose memory banks were severely damaged in battle with Superman in the film. Lex also seems to play coy with Brainiac in my opinion.
It's also confirmed by Perry White that Brainiac has caused trouble before, which places this film after it's introduction in "Superman: The Animated Series".
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Brainiac just hijacked Lexlabs. If he's gonna cripple Metropolis by crashing all our computers like last time..."
Lex Luthor recently acquired another small piece of Kryptonite after spending "two years and three billion dollars mining deep space", which he has kept in his possession as protection against Superman. In "Justice League" it's revealed that Lex carried this for years, and ended up getting cancer from it's radiation.
Now to explain Lex's attitude shift. Lex Luthor is un-characteristically in great spirits at Superman's recent failures (See the "Legacy" two parter) – but makes mention that as Superman's popularity rises, his falls. I agree it's jarring and this is the best explanation I can come up with.
When Phantom Lois mentions "That Green Lantern guy?", she's either referring to Kyle Rayner, whom Superman has worked with before, or John Stewart, who is active by this time (2002 in my timeline), but hasn't been seen on-screen yet. Most likely Kyle Rayner, since Superman has worked with him before.