I’ve run a 2.5” wt Maxxis DHF on a 25mm rim without any dramas before.
I’ve just swapped that DHF (which has recently been on a 30mm rim) for a 2.6” Spec Hillbilly and size wise it doesn’t look massively bigger – and most of the size looks like it’s in the knobbles rather than the casing.
So you might be ok – just the tyre will be rounder than it’s intended to be. Whether that’s a problem with that tyre design I don’t know.
I generally prefer a rounder profile as it’s more predictable when it runs out of grip / I run out of talent.
Edit – what fork is that pictured – it’s not a 32mm forked Rockshox fork is it? If so (and it’s non boost) you might be a bit tight on clearance of the fork. My hillbilly is ok through a non boost Yari though so maybe you’ll be ok.