You would be irresponsible to stay with this loser.
You and your DC (whether they're his or whatever) deserve better, I speak as the child of an alcoholic, I WISH my mother had left him instead we were subjected to all the crap that comes with addiction in the family.
It is NO life for any child BELIEVE ME.
Tell your sister and anyone else relevant the truth, stop defending and protecting this selfish arsehole!
Move into the new place without him and bin him off.
He clearly isn't really ready to get sober.
How long had he supposedly been sober for? It doesn't sound like very long.
Even IF someone chooses to stay with an addict the general advice is for the addict to live elsewhere if possible while getting sober and for their partner to only CONSIDER taking them back if there is long term recovery already in place.
It sounds as if his abstinence was recent and very short lived and only done because you gave him an ultimatum?
Addicts don't get better until THEY want to and even then it's not guaranteed.