Update: a bit more good news:
We had a court date yesterday, at which the judge signed an order amending our divorce agreement to establishes that Kid lives with me full time. So now it is as legal as it could possibly be. The judge also signed an order entitling me to full child support going forward and to child support arrears to be paid off at so many dollars per month. There is still disagreement as to when the clock started ticking for arrears, so we go back to court in a couple of weeks, I believe that that will not be a huge deal, as the judge accepted in principle that I was entitled to full child support while Kid was living with me full time, so what remains for me to do is to show that Kid really was living with me full time the whole time, which is not difficult to prove.
Ex did not show up (again) for court. His lawyer, who represented him in court, also can't find him. I suspect he is at the bottom of a bottle. His lawyer essentially caved, saying "okay, we'll consent to this" to the items on my "wish list". My lawyer was excellent - calm and low-key and very pragmatic.
The child support order is enforceable by local agencies - I expect that actually collecting the $$ will be a challenge for them, especially if ex gets fired (he's been off work for two years) but it's in their hands, not mine. They have the power to garnish wages, put liens on property and vehicles, seize financial assets etc - which doesn't help a whole lot with someone who no longer owns property or a vehicle, may not be employed for much longer, and has blown through all his savings. But in any case, it's not my job to chase after him.
My advice here would be
a) persistence - keep your eye on the goal. In my case, it was legal recognition that Kid lives with me and that her access to her father is dependent on SoberLink - the child support was secondary;
b) find the best, calmest lawyer you can. The more your lawyer resembles a Vulcan, the better;
c) don't be distracted any more than you have to be by the day-to-day freakouts of the alcoholic - this is a long game;
d) document document document everything you can and make sure your lawyer knows what you've got (one factor in my good outcome was that ex's lawyer knew that I had hundreds of pages of documentation and correspondence going back several years that demonstrated ex's problems with addiction and mental illness, and that by settling my claim rather than going to trial, all this documentation would stay out of the public record);
e) remember that the only thing anyone cares about is the well-being of the child - so all other drunken antics or bad behavior are irrelevant unless they bear directly on the child's safety.
I will continue to keep you all posted, and I really appreciate the support and interest that people have shown on this board. I'm especially thinking of CentralOhioDad, CoParentToA and baw81, who are all in this boat too.