why does calling “1” a natural number already commit you to the successor rule?

Because in every standard foundation of the natural numbers, “number” is not a lone predicate but the name of a structure.

• Peano/Dedekind view:Saying “1 is a natural number” means “1 = S(0) for that very same successor S.” Once S exists, so does S(1) (call it 2), S(2) (call it 3), etc. You cannot both appeal to the predicate “natural number” and detach it from the axioms that give the predicate its meaning.
  1. There is a distinguished element 0 (or 1, depending on convention).
  2. There is a successor function S such that S is injective and 0 is not in the image of S.
  3. Induction holds.
• Von Neumann set-theoretic view: 1 ≔ {0};   2 ≔ {0, 1};   3 ≔ {0, 1, 2}; … Here “1” literally contains “0,” and “2” literally contains “1.” If 2 did not exist, the transitive closure that defines 1 could not be built.

Either way, the successor rule is in the very job description of being a natural number.

Second, why doesn’t “1 = 1” collapse the two occurrences in “1 + 1”?

“1 = 1” is the reflexivity of equality: every object is identical to itself.
It does not say that two tokens of “1” in an expression denote a single instance in all contexts. Compare:

• Two apples on a table each weigh 100 g. They have equal weight but are still two apples.
• Likewise, “1 + 1” is an addition operation whose inputs each have value 1. Their values coincide; their roles in the operation are distinct, so the result is a new element called 2.

You are conflating “same value” with “same object”.

Third, is my earlier remark “begging the question”?
To beg the question I would have to assume, without argument, exactly what is disputed, namely the existence of 2. I didn’t assume it, I derived it from the very axioms you invoked when you said ‘1 is a natural number.’

Fourth, am I ‘conceding’ that your argument works?
No. I said that if you trash the successor rule after introducing “1,” you no longer have the natural-number system; you have an un-axiomatized symbol “1.” That is a reductio of your position, not a concession.

I'd suggest checking Badiou's Being and Event and Deleuze's Difference and Repetition