The most proper would be "Show me what you have".
"Show me what you've got" is more colloquial but also correct.
so "Show me what you got" is just shorter.
It's "What" because the thing is completely unknown, so we are using an interrogative pronoun ('what') to ask about it. You often hear "show me what", "show me how", "show me where".... show me + question word.
An example of "show me that" would be "show me that you love me". The fact ("you love me") is known, except for whether it's true or not.