Chase allows you to choose the Due Date, which determines the Closing Date. There is no need to call, this can be done online. Personal and Business cards have different offsets between Closing Date and Due Date. I have our personal cards set with Due Date of the 28th, and our Business cards set with a Due Date of the 25th. This causes all of the cards to Close on the 3rd. This reliably picks up all transactions from previous calendar month.
You can only choose a Due Date 1 - 28. (29, 30, 31 are invalid).
Not sure I understand why having a closing date of the third picking up all of the transactions is ideal for you?