In Amortized Analysis, can we chose how big $n$ is?
It depends a bit on what you mean with "choose an $n$". If you pick a fixed $n$, say a million, then no, that's not enough. Asymptotic notation already means "for sufficiently large inputs, the runtime is...". So a counterexample of any fixed size won't do. 

If however you show a method that for an infinite number of sufficiently large $n$ produces a counterexample, then this is valid.