Stepchild to use my LAST NAME different from LEGAL LAST NAME

I married a woman in foreign country who has a kid. I am US Citizen. Their greencard will come next year. They are here in US living with me on marriage visa. I have NOT adopted the child since that's a long process to be done in foreign country. Mom has 100% custody of the child so child will live with us 100% and biological father is untraceable. I am the Dad for all practical purposes and immigration purposes. My wife changed her last name after marriage to match my family last name. My own kid from prior marriage has my family last name. Quetsion 1) I WISH my stepchild to have the MY FAMILY last name as well for consistency in School, Insurance, Tax, SSN & Doctor with my family. HOW DO I DO IT ? Only child's foreign passport/greencard will continue to have original last name Someone said I can write child's name as "Xyz MyLastName (aka Xyz OrgLastName). Someone said I can write child's name as "Xyz MyLastName" as long as the intent is not fradulent. Child is minor under age 10. Someone said as soon as child gets greencard he/she can immediately claim US citizenship being a minor and being a child whose father (for immigration matter) is US citizen without going through naturalization process to become us citizen. At that time I can change his/her name to whatever he/she wants.