I don't even know how it would work but would love to see Padme secretly aiding the Rebels while raising her daughter. As for why she keeps Leia and gives Luke to Obi-Wan to hide? Simple - Luke's midichlorian count is much higher than Leia's and she knows the Chancellor has a strange habit of underestimating the strength of women.

Bonus - Vaderkin doesn't acknowledge Padme or the daughter he thinks is Obi-Wan's. Padme does nothing more to correct his misunderstanding, hoping to keep him as far away from Leia as possible.

Bonus Bonus - That whole 'I am your father' business comes with a side order of 'Luke, I'm your mother and I'm telling you to go to Dagobah!'