I heard from an attorney in Texas, if I am paying support, the child will have my last name. I hope this is true...
You aren't married to the mother, chances are you won't be in the hospital room when the birth cert and Social Security Card application are completed. The name of the child should not be a point of contention or distract you from your primary mission.
In the grand scheme of things, your child could be named Englebert Lipschitz-Humperdink and it won't mean a damn thing, if you are able to successfully fight for and enforce your right to be a father to your child. The child will only have to live with a name for 18 years...after that, it's the kid's business and easily changed by court order. And if it's a daughter, her last name may be fluid over the course of her life, anyway.
There are three things worth focusing on:
Making sure you are able to confirm that you are the father via DNA (don't you dare sign a paternity acknowledgment!).
Making sure you secure and enforce your rights as a father if the child is yours.
Making sure you are doing all you can to improve your earning capacity and ability to provide a safe, stable and emotionally healthy home for all of your children during your parenting time.
~
One other thing: she has two other children...so do you. Don't ever attempt to disparage her character based on her having other children. Here's why:
You knew she had children before you impregnated her (accident, schmaccident). You knew she was receiving CS for those children. If she's [XYZ] for having other children and receiving child support, you are at minimum a horrifically bad judge of character for getting involved with her.