Prior to what you said it was similar case I had, my best advise is allow your wife to work and you do the babysitting thats all I can say but about that 3 months that your wife have to work before you can apply I can't really say because what they need is that your sponsor which is an eea citizen have to be actively working by providing proofs work contract, payslips, bank statements that shows her salary on it and so on... I know it's compulsory your eea citizen have to do her treaty right in the UK also check the eea2 form to find out what else you can do or use to apply