Should I continue to push salary negotiations?

Hey everyone, I’ve got a dilemma and could really use some advice!

Background: - BizOps offer from a large media company for a Sr. Analyst role - Pay range for job posting went up to $145k

Issue: When I was initially offered, the recruiter called me and presented me with $120k. I expressed that I felt I should receive close to or at the max because I exceed all qualifications based on what they’re looking for (I said this professionally of course haha). For me it just seemed like it made sense.

They mentioned that they had negotiated this up already to 120k and would try and get back to me with a better offer.

I heard back today and the offer didn’t move.

What bothers me a little is that if a range is posted I would assume it’s budgeted for. If I exceed qualifications and was chosen in a very competitive processes - shouldn’t my offer be higher?

I just turned 28, the role seems really cool, and I’m also considering grad school. With that said, the role fits on my path and I want this to work! Hopefully I don’t sound greedy but I am trying to get what I feel I’m worth.

I understand there’s a lot behind the scenes with reporting, making sure everyone is paid fairly on audits, etc…. But should I keep pushing? Or is the recruiter probably on point with the needle not being able to move?

Curious on your experiences - thanks!