How many back and forths is acceptable in salary negotiations?
I've only ever negotiated two salaries and they went like this:
HR: What are your salary expectations?
Me: Based on this position and my experience, I believe a fair compensation is $X0,000
HR: We would like to offer you the position but at $Y0,000
Me: I am really looking forward to working with you. However, I do feel the offer is below the value I bring to the team. I am looking for the $X0,000 range
HR: [Day or two later] We can offer you [somewhere between $X0,000 and $Y0,000]
Me: Great, I accept.
However this time I have a problem. I was offered a government job that has a set salary range. I thought they would offer me the higher end of the salary (I have almost double the experience requested and Masters when only Bachelors was required) but it was more toward the lower mid range. When I requested more, they suggested that it likely wouldn't be approved because the request would go outside their department and those types of requests never get approved. They implied they would try but not to expect the approval.
I really want this job but they are offering 15-20% less than I was expecting/hoping for. If they come back with just a minor bump, how hard should I push? I decided I will probably take the job regardless because of great benefits, but it will be a significant impact on my long term earning potential since I plan to be here for at least 5 years with only meager annual raises. I don't want to start off on the wrong foot but I am worth it gosh dang it!
TLDR; How many back and forths is acceptable in salary negotiations? Would two rebuttals be too rude?