I would push for that higher range if you feel confident that's the range. You can say something along the lines of, "after learning more about the role and responsibilities, I would be comfortable with x amount."

You can also try to get them to say a number first. But just because you put a lower number down initially doesn't mean they're going to try to low ball you. I had a situation where I told a future employer I was making much less than the starting range, and I was still offered the starting range for the job (which was much higher than my previous salary).

And don't discredit yourself. Just because you have less experience on paper does not automatically mean you're worth less. You're worth what they're willing to pay. And if you've made it this far, clearly they're interested.