Or...
If Neil is recruited in the future (which is never explicitly stated) he knows the full events as they transpired and therefore also knows that Kat is a vital piece in completing the mission, ergo why he steers JDW into that direction with subtle clues.
Also, I don't remember the kid ever being referred to as Maximilien but I welcome any evidence as seen/heard in the actual film.
It was explicitly stated he was recruited from the future version of the protagonist. Neil outright told the protagonist that. So, the Neil we see is from the version and he inverted himself back through time to complete the mission. But there were plenty of inverted characters that were part of the TENET organization, and the main rule of TENET is not to pass on information. Everything was from a need to know. That's why throughout the movie, everyone the protagonist talked to, only could tell them just what THEY know and nothing more. Neil pretended to do that as well, but he's the ONLY one that really knows the full picture, to the point of knowing that he had to go back in the turn-style to sacrifice his life for everything to work out as it was destined to happen. So why the future version of the protagonist inform him everything and not anyone else on his team? What made Neil SO SPECIAL over anyone else of the TENET organization to be privileged to the entire information? If Tenet is consistent as an organization, he wouldn't have known what his purpose was, only that he was to be recruited by the protagonist to assist the protagonist's younger self. He would have to actually known everything from the beginning.
Also, why would Neil go out of his way, to even at the Opera house to protect the protagonist, unless he's personally invested? Max is someone that we were inform is highly intelligent as a child, and Neil mentioned about how he mastered the science behind physics. There's too much similarities and connections why is why it's been a popular theory throughout the internet.
And the name Maximilien is shown in the credits as an Easter egg.
One more thing to add, Robert Pattinson dyed his hair a more blonde color to which matches that of Max. There's no rhyme of reason why he would of dyed his hair for the role unless to suggest to look like the older version of Max.