Statute of limitations for lying on a birth certificate

Some background

I'm M42 here, and my father is not listed on my birth certificate. Parents give directly oppositional reasons as to why; mom says dad didn't want to be on the birth certificate and dad counters that mom didn't want him on it. Over the years I've come to believe mom, simply because of my father's lack of participation in my life. I know he loves me, but he got to be the fun-time parent while having no real responsibilities. Mom never talked poorly about him, but instead let me come to my own conclusions. Something which was not easy for her I'm sure.

Mom later told me that she was strong-armed into lying because my father was abusive. So there's an argument for duress there.

Mom is still paranoid that the government will come after her if they find out about this. She's in her early 60s and on disability. So, it's not like she has anything to take.

Why it matters now

I was named after my father (first name only, mother's last name) and I've never liked it. Over the last few years having the name of someone who wanted me named after him, but didn't want me enough to claim me legally has been bothering me, so I've been looking into legally changing my name.

Could this cause issues for my mom?

I was born in Santa Barbara County, CA, though I currently live in South Carolina.