The following is just my opinion:

Integration can be done with its own special arithmetic: Interval arithmetic. I base this suggestion on the fact that this is apparently the only way of automatically getting error bounds on integrals. It's cool that it works.

NSA does not work with a computable field so it's not directly useful. But at the end of the article, there's a link to some code that uses the Levi-Civita field, which is a "nice" approximation to NSA because it's computable and still real-closed. You might be able to do an "auto-limit" using it, in a kind of generalisation of automatic differentiation. This might for instance turn one numerical algorithm, like Householder QR, into another one, like Gaussian elimination, by taking an appropriate limit.

I don't know if these two things interact well in practice: Levi-Civita for algebraic limits and interval arithmetic for integrals. They might! This might suggest rather provocatively that integration is only clumsily interpreted as a limit of some function. Finally tbh, I'm not sure if this is the best solution to the friction/collision detection problem you're describing.