> professional painters will generally paint the place for much less than $10,000 you should be able to do this

Unfortunately, lead paint remediation will almost certainly cost more than $10,000 unless it's a tiny house. You can find unlicensed operations that will charge less, but then you risk the very large fines for hiring an unlicensed operation for hazardous lead paint remediation. I wouldn't risk it since the city is already looking into the situation.

But I do agree that you should be able to get an extension if you can show a good faith effort at addressing the problem, unless your city health department is totally unreasonable (always possible, I suppose).