That cases are both possible but have a different sense.

на + accusative = to, towards
на + prepositional = on, at, during
This alone perfectly explains a lot of my confusion.


Я пригласил её в кафе на нашем втором свидании. - Prepositional.

I invited her to the coffee shop during our second date. (I.e. for the third date.)


Я пригласил её в кафе на наше второе свидание. - Accusative.

I invited her to the coffee shop for our second date.
That perfectly explains it well. So is my corrected phrase is impossible in this case:


Я взяла её в кафе на нашем втором свидание. It wouldn't make a whole lot of sense since it uses both accusative and prepositional.

Moreover, I'm compelled to ask why did you change the word взял to пригласил? It wouldn't be correct translation since the original phrase says "took" not "invited".