Initially, he wanted to ask her for this kind of help because she was a non-Federation citizen. Then, when he discovered who she was, he thought it would be even easier to do that. I suppose he thought it would be easier to make her mess up with Rathosians politics than ask her to perform whole "rescue the sun" operation, as the latter required much more work.
But she knows she can't have her children's love back, if she keeps repeating the same mistakes and wrong things she'd done. As you can see, in each of her "visions" they are still angry with her. Very angry.
In a Cardassian library
or in a Cardassian gallery
"Reagan, it appears, is really only an ardent unionist if the unions in question are in Poland" - Stephen King, Skeleton Crew