repeating S n times isn't something we know how to do in first-order logic

Why not? Repeating S n times is just addition, and addition is defined in the peano first order logic axioms. I just took these from my textbook:

∀y.plus(0,y,y)

∀x.∀y.∀z.(plus(x,y,z) ⇒ plus(s(x),y,s(z)))

∀x.∀y.∀z.∀w.(plus(x,y,z) ∧ ¬same(z,w) ⇒ ¬plus(x,y,w))

I've also seen addition defined recursively somehow, so each step it subtracted 1 from the second number and added 1 to the first number, until the second number was equal to zero. Something like this:

∀x.∀y.∀z.∀w.(plus(x,y,z) ⇒ plus(s(x),w,z) ∧ same(s(w),y))

From this you could define subtraction in a similar way, and then state that all numbers subtracted from themselves, must equal 0. This would rule out nonstandard numbers.