If the hyphenated name is on the birth certificate that is his legal last name. It can also be his legal last name if you put both names down on the birth certificate in the last name place. I think, but am not sure, that you have to send in a copy of his birth certificate to get a social security card. Thus his legal last name will be whatever name or names you put in the last name space of his birth certificate.
What you actually use for a name can be something shorter. You do have to be consistent or his records will get mixed up causing you and him when he's older difficulty getting complete records. You'll have to list all the names he's used as aka on a request for a copy of records. For example if he changes schools and for medical records. aka is short for also known as.
My daughter married and decided to use a hyphenated version of her daughter's name. this was a daughter from a previous relationship and had his last name on the birth certificate. She just used the hyphenated name and did not change it legally. Thus at school she had a hyphenated name on the roll sheets. But I noticed that her report cards came home without the hyphen and I didn't know which was the last name they used for filing papers. Eventually I noticed that they maintained her legal last name beginning letter for filing.
My daughter divorced and remarried and they dropping the last name of her previous husband and left my granddaughters name as it is on the birth certificate.
This example shows you how complicated it can get. Based on my experience I suggest that if you want both names to be his legal last name that you hyphenate it and always use the whole hyphenated name.
My grandson has two middle names. Forms seldom have room to write both names and so my daughter has been using both initials in the middle name slot or squeezes both names in. I noticed that on his school records they only list his first middle name.
You can do what ever you want with the names. If you'd like his legal last name to be hyphenated then fill out the birth certificate that way. Then decide how you want his medical and school records to be filed and always use that name. You can change your mind later and begin using a different sequence as long as you keep in mind how his records will be filed so that they can be retrieved.
I'm a retired police officer and in my experience many people use their names differently. Sometime hyphenated and sometimes not. Sometimes mother's last name and sometimes fathers. Children who come from Mexico nearly always use their mother's last name but once they get started here, come officials insist on using only father's last name or a combination of both. Sometimes mother's name first and sometimes father's name first. It can get confusing but it's still OK, however you do it. That is why on official documents there is a space for aka or the
I have a friend who married and she legally changed her name to a hyphenated one. when she was in the hospital for the birth of both of her children the printed admittance record only showed her maiden name which was the first name when hyphenated because that was all the space available. The receptionist did find her but she was confused and wasn't sure the name was for the person I wanted to see. lol
I rather like the idea of being different and suggest that you go for a hyphenated name if that is what you want to do. Both my first and last name are unusual and I haven't minded having to pronounce and spell my name for people.