I'd be skeptical of that assertion. Even sticking to ordinary topology on actual sets, transfinite sequences are not enough to do limits in general; in general you need nets. (Or filters.) Doesn't mean you'll need that here -- might the fact that the surreals are linearly ordered help? -- but I don't think it's something you should assume would work.

But yeah it does seem like you'll need something able to contain a "sequence" of order type that of the class of all ordinals; quantifying over ordinals or surreals or something in the "domain". (Like, as I said above, limits of surreal-valued functions of a surreal variable shouldn't pose a problem.)

In any case, sequences or nets are not necessarily the issue. This still doesn't help with infinite sums, because those are still just ordinary omega-sequences. But really the issue is integration; infinite sums can be ignored if you can get integration. Does the "domain" there have sufficient granularity? Well, uh, I don't know.