Which one sounds better?

紅茶2杯をお願いします。
I’d say:
紅茶を2杯お願いします。
2杯の紅茶をお願いします。

The first one is written on a flashcard I downloaded.
And having a particle after a counter is wrong. Am I understanding this correctly?

See a translation

Well, having a particle after a counter sounds a bit unnatural, though I wouldn't say it's wrong.
All of them are grammatically correct, but 紅茶を2杯お願いします。sounds the most natural between them.

By the way, it seems that you are just asking about grammatical issue, so just ignore the following if it isn't concerned with what you want to know at all.

If you imagine a situation in which you are ordering two cups of tea for you and someone (e.g. one of your friends) in a cafe, more natural expressions are actually as follows:

staff : ご注文お伺いします。(May I take your order?)
customer : 2人とも紅茶で。/ 2人とも紅茶でお願いします。 / 紅茶2つ。 / 紅茶2つお願いします。

紅茶を2杯お願いします sounds as if you went into a cafe alone and ordered two cups of tea at once for yourself.