1) How did you get a green card? You may not need to file an I-130 for your son depending on how you got a green card.
2) Custody does not matter for the green card itself - if the child was born during a marriage then all is good.
3) However, the child cant just move to the US alone! Custody does not matter - but you have no permission to bring the child here.So better wait until child turns 18 - as long as child under 21, you are good. (when i say wait, i mean
when the national visa center contacts you, you can delay replying and ask for an interview after son turns 18. Otherwise the custody issue will come upduring the interview.)
Even if mother has custody, she can sign a paper that she allows son to go to US.
Just like ppl go to high school in different town.You dont need formal custody for this issue.Formal custody only needed for citizenship.
In conclusion: 绿卡不需要办理监护人转移手续。等你儿子到了18岁,再来美国。
For details, read INA 101(b) for the definition of a child.
