Generally the courts will side with the best interests of the child. If there is proof of alcoholism inhibiting the father's ability to care for the child, then I can't see how a judge will grant 50/50 custody.
It's also worth mentioning that child care costs have only a small part to do with how much time a parent spends with the child, but it's more based off maintaining the same quality of life between the two households. In the case of my friend, he actually recieves child support because the ex wife makes millions per year while he struggles to make ends meet.
I would also owe my husband spousal support and pay high child support even with a 50/50 parenting time split.
In this case, definitely raise the alcoholism issue. Breastfeeding as well, if it applies.