meet you at lunch?in lunch time??

What are you doing at lunchtime? As a bonus, I was allowed to swim during lunchtime. I would like to know if it is correct to write at lunch time (two words) as I learned that we usually write 'lunchtime' with one word only.

I will meet you at lunch
I will meet you at lunch time

Both are fine... The first one is more colloquial.
It's always "at lunch," not "in lunch."

Or, I will meet up with you for lunch.

prefer "we'll meet at lunch break".

Both are wrong.

Correct sentences:
I will meet you for lunch.
I will meet you at lunch time.