Learn English – What does ‘The Thick End Of’ mean, and why

Why here is an interjection, placed at the beginning of a sentence to express surprise:

Is it nine o'clock already? Why, I must have fallen asleep!

or opposition:

Are you suggesting he stole the money? Why, I think that's impossible.

or in you example, emphasis:

B. Awesome, thanks!
A. Why [would you think it would be any less than awesome?], yes of course.

or perhaps

A. Why [would I even need to be thanked for something I'm happy to do], yes, of course.

Don't take the bracketed words as a literal ellipsis. The why is there to express a general emphatic tone.

The OED finds the interjectory use of why going back five hundred years.

My suspicion is that your conversation partner wants to have a catch-up, rather than a catch, i.e. a conversation in which you catch up:

4. To bring (another) up to date; brief

   Let me catch you up on all the gossip.

[AHD] While this sense does not seem to be found in dictionaries, examples abound online, e.g. “we have a catch up with Neil Hamilton” or “after the game I'm sure we'll have a catch up and a bit of a yarn”.

But if they did indeed say have a catch, it is a metaphorical usage.

To have a catch is to play catch or to have a game of gatch, in the first noun definition of catch as given by AHD:

1b. A game of throwing and catching a ball.

In my personal experience, the phrasing have a catch is mostly Northeastern, though I cannot present evidence that it is a regionalism.

The archetypical game of catch, for Americans, is a father and son tossing a baseball back and forth— providing an excuse to spend time together and have a conversation. Thus, the teary closing lines of the 1988 film Field of Dreams:

Ray: Hey... Dad? You wanna have a catch?
John: I'd like that

Thus, if the other party does want to have a catch with you, it means they want to do some activity together where you can have a conversation, perhaps because conversation by itself would be less worthy of time, or perhaps awkward.