Jeff, Jay & John seem to have it down pretty darn well: find the hot button and exploit it It's really not devious because this is for the greater good and will help your family in the long run, right?
However, let me caution you that using romance as a tool to get what you want is very shaky ground and can land you in heaps of trouble. It's a subtle form of manipulation (and never mind that women use sex to get men to do what they want us to. For some reason, they always get a pass with that one)
I think laying out the facts on a list and showing what you will do to offset the space issue and perhaps the money issue is the best policy. Plus, it shows her respect.
If she doesn't see eye to eye with you after all that, buy it anyway
Good luck!
Davy