Btw one thing worth noting if you really do want to work with surreals is that it may be more productive to think in terms of { stuff | stuff } rather than limits. (Similar to my “break ties by least birthday” suggestion.) Sequences don’t have limits in the surreals, but there is nonetheless a theory of surreal exponentiation based on {stuff | stuff}. Integration… well, it’s less obvious to me that integration based on limits should fail, but if it does, you could try to do it based on {stuff | stuff}. (The existing failed attempt at a theory of surreal integration takes that approach, though as I said above, that’s not really the same thing, as that’s for functions with the surreals as the domain.)
Btw one thing worth noting if you really do want to work with surreals is that it may be more productive to think in terms of { stuff | stuff } rather than limits. (Similar to my “break ties by least birthday” suggestion.) Sequences don’t have limits in the surreals, but there is nonetheless a theory of surreal exponentiation based on {stuff | stuff}. Integration… well, it’s less obvious to me that integration based on limits should fail, but if it does, you could try to do it based on {stuff | stuff}. (The existing failed attempt at a theory of surreal integration takes that approach, though as I said above, that’s not really the same thing, as that’s for functions with the surreals as the domain.)