When it comes to “getting the job”, the specific degree path doesn’t matter. The employer just wants to know if you can solve their problems or not. It all boils down to what you want in your career. Software engineering is a small subset of computer science so if you majored in Software Engineering, you’d be studying a highly detailed section of Computer Science.
Career wise, this seems like the way to go. You want to be a software engineer, majoring in Software Engineering sounds about right. There’s a catch 22 here though. There are concepts in the Computer Science degree path that fundamental to being a “good” software engineer. You can take the time to learn these things on your own, sure. But by majoring in Computer Science, you become a more well-rounded individual that can solve problems in completely unique and satisfying ways. So my vote is, Computer Science.