I didn't say easier or faster.
The father, although he isn't in her life, could object.
You haven't known this current guy too long.
I'd wait five (or more) years.
I'd at least until I'd been married for five years, before even considering changing her name. When you decide to do so, go forward. You'll discover changing a name to be an expensive and lengthy process.
In the end, does it matter more what her name is, or that she becomes an educated, well behaved person?
The name change means more to you. The child doesn't care. Your kids just wanna be loved and know they are safe!!!
Stop beating yourself up over what happened in the past.
Just be the best woman and mommy you can be going forward. You do that, and love your babies (not some lying, irresponsible man-thug), and all of you will have a great life.