As P.E. Dant mentions in his comment, the phrase is just a colloquial variation on "Ask me whose invitation I have". But let's break it down.
In English, the verb "to get" is used in a wide variety of ways, often with subtle differences. For example:
"I got $100" = I received $100.
"I have got $100" = I possess (or I am holding) $100
Note the difference -- I receive vs. I have something. It would not be redundant to say that I get something, and then I've got it. Some other examples:
I got your invitation.
I've got your invitation.
Whose invitation did I get?
Whose invitation have I got?
In the movie, Brendan is not saying he received the invitation. He's saying that he has the invitation, and possibly that he's actually holding the invitation at that moment.
The second structure, "Ask X if Y" is an alternate way to phrase a question as a kind of imperative
Do you want to know if I like chocolate?
Ask me if I like chocolate.
Does she like chocolate?
Ask her if she likes chocolate.
You will generally use this structure to imply that you already know the answer -- that it's a rhetorical question -- but you want the person to ask anyway:
A: Ask me if I love you.
B: Do you love me?
A: Yes, of course! I love you more than pizza, and that's saying something.
Put the pieces together, and you have Brendan telling Laura that he has an invitation from someone, and he thinks she will want to know about it, so he tells her to ask him about it.
As a side note: The movie Brick is in a style known as "hard-boiled detective fiction", with the twist that it's set in a high school with teenagers as the main characters. If the dialogue seems oddly adult for such young people, it's because high school students do not normally talk this way -- the setting is a deliberate choice by the filmmaker to add a surreal element to the movie.