For weeks viewers of Live With Regis and Kathie Lee, the nationally syndicated show, have been listening to Regis Philbin's unabashedly perky cohost, Kathie Lee Gifford, go on about her two-week singing stint at New York's swanky Rainbow and Stars cabaret. The hype must have helped, because all 20 performances sold out despite a $35 cover charge and a pricey menu. ''I can't rap, I just can't,'' Gifford giggled to her nightclub audience. Instead, not altogether surprisingly, she performed a medley of ''my favorite happy songs,'' along with standard oldies and show tunes. But there were some surprises: One night Rich Little did a 10-minute impression of George Bush; on another, Lee's obstetrician showed up ringside. ''It felt funny having my clothes on in front of him,'' says Gifford, who gave birth to son Cody last year, ''but I said, 'Thank you for the 16 hours of labor. Thank you for the cesarean,' and I asked him, 'How many centimeters dilated was I?''' Now that's grace under pressure.