Brooke Burke and David Charvet
Jen Lowery/Startraks
"We've waiting our whole life for a son, and we haven't named him," Burke told PEOPLE at Tuesday's Architectural Digest Harry Benson photography retrospective in West Hollywood. "He did leave the hospital without the birth certificate. I know that's terrible!"
Now, nearly three weeks after giving birth, the couple may be finally zeroing in on the perfect moniker. "We're really, really close [to a name]," said the 36-year-old. "We've agreed on the first and the last, and we're still fighting on the middle."
The newborn joins the couple's daughter, Rain, 1, and Burke's two girls – Neriah, 7, and Sierra, 5 – from a previous marriage.
"Going from two to three was easy. Going from three to four, a whole other world," a weary-eyed Burke said at Tuesday's event, where she and the Baywatch alum enjoyed a date-night out.
In fact, admitted, "We're so sleep-deprived right now, we can both barely see straight."
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