Question: Does Kat ever wise up again to Kara's secret? That and why did she (Flockhart) leave the show only to come back?
Question: Why did they move the show to Vancouver?
Answer: For one, it's usually cheaper to film in Vancouver because of the tax breaks. But all the CW Arrowverse shows (with the exception of "Black Lighting") are/were filmed in Vancouver. So it makes it easier and cheaper for sharing set designs, crossovers, etc ("Black Lightning" filmed in Vancouver during its crossover). Originally "Supergirl" wasn't part of the Arrowverse when it was on CBS, but it was too expensive to produce and wasn't reaching the wider audience the CW shows had. So CBS moved it to its sister company, The CW.
Answer: Yes, s02e22. As Kara leaves her office, she said "go get them, Supergirl" to herself. After season 1, the show switched from a CBS production to a CW production and filming moved from L.A. to Vancouver. Flockhart isn't really a fan of traveling to shoot, and didn't want to be away from her family for so long.
Bishop73