Or do I just go with my old resume and hope that my new job doesn't appear on any potential background check?
No, don't do this. It's fine that you took a new job, and it's fine if you leave only after 2.5 months. It sounds like you're young, and moving around is expected. Now if you change jobs every 3 months for the next 3 years, that would be a red flag.
Just say the new job would be a better fit, or better future potential, whatever you want to say. It's not a big deal.
You can update your resume with your new job, but I wouldn't give it to them unless they ask for it. The experience of your current job might even help you at the interview.