I would start steering the conversation away from career levels since the impasse here is getting stuck on salary bands by level. It's a bit difficult when there are rigid corporate policies in place, but as a hiring manager, I know there is room to extend unique offers for top talent. For example, at Director+ level the "best" offer vs. the "standard" offer can as much as 2X, but the concept still applies to level below as well.
Articulate what you bring to the table and the value that unlocks for the company, especially if it is unique vs. most other candidates. Here are a few examples.
Depending on your relationship with the hiring manager and rapport built during your interviews, you can consider dropping them a message. It's hard to advise how to do this tactfully without more context, but most of my previous jobs has been a verbal with my future manager and the recruiter is only there to complete the process.
Assuming that the offer is genuinely at the top range, can I still attempt to negotiate given that the team believed I'm suitable for a staff SWE uplevelling?
You can, but I would only take just 1 crack at it and leave it at that. Just make sure that crack is a good one:
If they really can't budge on the level or TC, you can ask for a signing bonus. That's generally easier to be approved as that's a one-time payment as opposed to recurring.
Another thing I will say is that while 10% is substantial, it's not the end of the world to miss out on it. It's much better to be on the rocket ship that's accelerating quickly instead of one that's already high in the sky but stalling out. If you genuinely believe this team has a wonderful environment for you to grow and be passionate about your work, the pay cut could be the right call.
I recommend these resources on negotiation as well: