I am honestly kinda confused as to why people equate fantastical elements to "campiness and silliness". Some of the darkest Batman stories have been extremely fantastical.
I keep saying that tone has very very little to do with why these two universes are not compatible at all.
“Tone” or “tonal compatibility” are slippery and ambiguous concepts. Moreover, and as you mentioned, they’re sometimes conflated with the realism vs. fantasy issue.
By way of anchoring these concepts, I would offer an extreme hypothetical that (I
think) most would agree with: Pairing Pattinson’s Batman with Frank Gorshin’s Ridder (from the Adam West TV series) would be a
bad idea.

Why? Well, they’re from
different genres,
different approaches to the characters and with
different creative goals for the respective movies. One is a serious serial-killer mystery, the other a fun/campy adventure. Now as a shorthand, I’m fine with calling this difference and incompatibility a matter of “tone.” But if there’s a better word, I’m happy to use that instead.
So… to the thread topic: Would Corenswet’s Superman be compatible with Pattinson’s Batman? Obviously, we’d need to see Gunn’s
Superman to make a proper assessment. But if Gunn’s approach is close/adjacent to the Donner/Reeve interpretation, my answer would be
no. The
“tones” [insert preferred term] would be too discordant. And forcing each into the other’s genre context (or some middle ground) would be a disservice to both.