I could have sworn that I have seen surreal integrals calculated as part of research into surreal mathematics. To me surreal calculus is a thing.

There's a well-known attempt to make a theory of surreal integration; it produced some fruit but did not actually yield a sensible definition of surreal integration. I'm unaware of any successful attempt.

Edit: Also, that was for functions from surreals to surreals, not for functions from a measure space to surreals.

Are you sure you are not confusing how infinities are handled in other formalizations? Surreal addition is well defined and it takes no special form in the infinite range.

I'm not disputing that? The (or rather, a) problem is infinite sums (sums of infinitely many things), not sums of things that are infinite.

The sentence structure seems to suggest having a proof that such things are not possible but I am kinda getting the situation is more that you lack any proof that it is possible.

I speaking weakly since I didn't really feel like dragging up the actual arguments. I'll expand on this in a cousin comment.