Honestly, I am very divided about how they handled the character: on one hand I can't deny they did something really well which No Way Home was kinda lacking: the classic tunes associated with the character.
It may not seem like much to some, but getting to hear the themes which many have grown up with both in the 90s with the original two Batman movies with Keaton and in the mid 2000s with the LEGO Batman trilogy and remixed or reorchestrated feels kinda amazing. It helps boost up scenes such as this one:
This whole scene is amazing because the music really carries It, ngl. If they used anything else I feel It wouldn't impact as much.
On the other hand... Is It just me or they made Keaton's Batman too epic and larger than life than how It actually was in the original movies?
I get that we are talking about one of the most iconic and legendary takes on the character, and as such the crew of the movie were excited to be working with him... But idk, in the original movies Batkeaton felt more like a character in the lines of someone such as Blade or say the Teenage Mutant Ninja Turtles (the original black & White comics version, not the cartoon ones). He killed, he used fire weapons (on his vehicles) and even smiled while doing so. I feel the version of The Flash ignored these faults on the philosophy of this version to instead show how epic and amazing this version was... Which imo isn't ideal.
No Way Home for example knew to take their characters and events from the Raimi and Webb movies seriously while also acknowledging the flaws or imperfections of said characters (Electro finding Connor's plan silly for example, or how Andrew Garfield couldn't forgive himself for letting Gwen die). I feel they should have done something similar with Keaton's Batman, and in fact one of my favourite moments in the whole movie (and the only joke I laughed at) was when they are on that elevator about to set an explosion and asks both Flashes how much they weight to set up the proper one. That was funny, and acknowledged the fact that Batkeaton was calculative and very researchful in the original movies.
Tl;dr: Keaton's Batman in The Flash succeeds on the music and sound department, but in philosophy It doesn't really treat the character with the honesty and awareness of the villains and Peter 2 and 3 in No Way Home