It's impossible to tell if you would pass a background check. Like others have suggested, if you do, or if there's none, and you get to an interview and end up being questioned, own it.
That being said, it's likely going for a small company will raise your chances. Especially one that isn't in tech. Software engineers are used in many companies, I'm not saying you will necessarily get the most thrilling job, but for instance, a book printing company could have software engineers, while the recruiter may not know much and may not care about Googling people.
The next things you may be allowed to do are:
- Change your name (legally).
- Use a nick name (I know people who have used a different first name their whole life, because they didn't like their given one, few people know their actual first name).
If you can land the job, prove your skills and professionalism, when the story comes out (possibly from you), you may be in a better position to keep the job.
Note that I'm answering from a "what can you do" point of view. I leave the ethical part up to you.