Be diplomatic about it first.

Get the intel (factual) that these other offices need people and have the spots for you to move in. Refer to them by position/location, even point to postings if listed on line.

Send these to him via email and refer to the initial denial in the email.

If he gives you ANY excuse afterwards that doesn't include inside knowledge on the roles (promised to other hardship cases, postings are outdated and those roles aren't open anymore as transfers have already been processed, etc.) then forward that chain to his manager and explain your situation. In that email, explain that you'd like to keep it private so that your current manager doesn't feel undermined and that you're just trying to do what's best for your family. If it's not possible, you understand but if it is, you'd very strongly like to be considered for the hardship case.

If that answer is no at that point, no harm, no foul. Your current manager doesn't grow further animosity at you and you can just look for a new gig while making more plausible current arrangements (I agree with Rex on the car, especially if wifey got a promotion. Should be a lil extra cake to afford it).

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