You absolutely can. Managers will push back with "rules" that only apply if management doesn't want to pay you more. Or they will go to HR to get an exception if they think you are worth that exception (that is, if they aren't worried about not being able to match an offer for an employee that they really care about). You can absolutely negotiate.
In the past, I've been quite open when I thought that I needed more money to my manager, and have even given specific ways of making me "not distracted by money concerns". Sometimes they can meet those goals, sometimes they can't.
Personally, the offer as you've given it is probably more adversarial than I'd prefer. Something like "I feel like I'm worth more to the company than X, I feel like I'm worth Y, and here is a list of reasons, here is my career goals, etc etc". Then if they don't match it, you can accept that other offer. But YMMV.
In the past, I've been quite open when I thought that I needed more money to my manager, and have even given specific ways of making me "not distracted by money concerns". Sometimes they can meet those goals, sometimes they can't.
Personally, the offer as you've given it is probably more adversarial than I'd prefer. Something like "I feel like I'm worth more to the company than X, I feel like I'm worth Y, and here is a list of reasons, here is my career goals, etc etc". Then if they don't match it, you can accept that other offer. But YMMV.