If all that you're saying is true, and if the father's home doesn't have major issues or threats, then I agree that it's probably best for the kids to ultimately reside primarily with the dad.
However, I think you should spend a bit more time thinking about how to make that happen, and it may not happen as quickly as you would like. You've already been shot down at an Ex Parte motion to change custody, so either your attorney royally sucked (or the dad did, if not represented) or the judge is not extremely alarmed at this point. That tells you that you need to build your case on solid evidence, not on speculation or character assessment.
First, if the mother had court orders to provide medical insurance for the children, and if you have evidence of those court orders, I would suggest that your attorney or you send a very business-like letter to the mother. The letter would essentially say, "As the $4,000 bill is a result only of the mother's willful contempt of orders by allowing the child's health insurance to lapse, father is willing only to be responsible only for what the co-pay would have been. According to the lapsed policy, an out-patient Emergency Room co-pay is $75." Unless father has failed to pay child support (and hence mother couldn't pay for insurance), I don't see a court holding him accountable for any amount above what the co-pay would have been.
If the mother took the kid to the ER, then it's her bill. You won't have to worry about bill collectors or a bad credit mark.
Second, if you have valid court orders stating that the mother shall not move more than 25 miles away from father, then you simply expect those orders to be followed. If the mother moves more than 25 miles away, then she's in contempt. She would have to petition the court to modify that clause, at which point you've got a move-away case on your hands. If she moves and ignores that court order, then you may be able to try for another custody change, following a contempt hearing.
In California, there was an important Supreme Court case called LaMusga that occurred in 2004. It clarifies that while custodial parents have the presumed right to relocate the kids, so too must the court rule in a child's best interest at all times (i.e., where the relocation may be presumed in the best interest, barring other evidence). I think that case would benefit your situation, if/when the mother tries for a move-away. You may, at that point, take all the information you outlined and argue that a custody evaluation is necessary to assess at which home the children shall primarily reside. Your basis will be that there is a change of circumstance as a result of the the CPS investigations (especially if any are found to be neglect), the irresponsibility demonstrated by the mother, and whatever else you gather along the way. The move-away in itself is not a change of circumstance, though it would be relevant to point out that the father would likely be stuck with the burden of transportation as a result of the mother's pitiful finances and suspended license.
As to the Section 8 housing fraud, just let her dig her own hole. You know it's all going to come tumbling down anyway, so I would suggest staying at an ethical place. You did write me a subsequent note (not posted), saying that you were re-thinking along what I'm suggesting anyway.
The biggest problem with your proposed argument is this: you're speculating based upon reason, not evidence. The court cares about evidence of things that are happening or have happened. The court won't entertain your speculation about future gloom and doom, because the court knows that the mother may inherit $100,000, or may win the lottery, or may get a job, or may find a sugar daddy... or may fall on her face. But the court won't make a decision until any of those things happen.
So, you bide your time, documenting everything, and building your case. I strongly recommend you drop seventy bucks on Win Your Child Custody War
by Hardwick. It's the most valuable thing you could have by your side right now, as it will guide you on how to gather relevant evidence to return to court and successfully argue what you already know is best for the kids. There's a link to learn more about the book on my Resources
page, and you can buy it at Amazon if you want.
The thing is... yes, it may be best for the kids to move to the dad's home RIGHT NOW. But unless they're being molested regularly, starved, and locked in cages at night; you'll really have to build your case appropriately (i.e., appropriate to family law) to effect a positive change for them.