Would that be the case here with regards to the Morse Theory tag, or is that a bit of a stretch?

I think the question is answered precisely by Morse's Lemma, hence my suggestion.

@MichaelAlbanese I do not know Morse lemma. So I am not able to make this call. But since you are familiar with the area and you posted the answer based on this lemma, I think you might simply go ahead.

For the benefit of the readers who will look into this room and are not familiar with the whole adding tags based on answer business, I will add link to the relevant discussion on meta: Retagging after an answer is given. (Personally I am in support of this policy and the voting the linked meta thread suggests that this should be considered consensus of the MSE community.)

In short, what I wanted to say, if this is topic you are familiar with, it seems reasonable to follow your best judgement. If somebody disagrees with the retag, they will probably let you know.