I have never seen version 2. anywhere and to be honest, it makes no sense as first it's hard to cover all the cases it creates with that way of interaction. If the data input field is the last element on the modal will the modal create a scroll area just to adapt a picker inside it? That would result in an extra, unexpected action from the user to scroll just to see the whole month.
What happens after you select the date in that scenario? Will the modal box resize again to hide the scroll?
I don't think we can find any test data on your questions regarding these scenarios but given existing design patterns and the extra cases and issues the interaction presented in case 2 might cause I think it's safe to say that case 1 is the go-to.