Similarly the natural numbers are a process that can produce more natural numbers, and can do so for however large a limit you choose.
The problem of rearranging the elements of a set to get a different sum is handled here because you are looking at the limit of the sum, not the sum of the abstract set. Just as in classical math you have to make sure that to compute { sum_i=0^inf (-1)^i / i } that you are actually computing { lim N->inf sum_i=0^N (-1)^i / i }, with intuitionist math you can't help but do the latter, because the former statement makes no sense except as the latter.
The "adding an item to a set makes it larger" is only problematic if you're trying to define "larger" for non-terminating sets, and in addition trying to treat them as "sets" divorced from their underlying structure. The rationals and the integers may look the same to a set theorist, but they are clearly very different objects and it's not at all clear what additional power you get from introducing morphisms that don't preserve any underlying structure.
But do you have something better? It’s not as if mathematicians immediately accepted this as the way forward. There was a long struggle accepting that lots of statements that are true about finite sets do not extend to infinite ones (examples: “adding an item to a set makes it larger”, “when summing a set of numbers, the result doesn’t depend on the order you do it”)
I think that you either have to accept this as the best way to treat infinite sets, or have to give up the notion of infinite sets, and that has its problems, too. For example, it would mean there’s a largest integer.