First off, I'm sorry you've having to deal with this.
It's exhausting and stressful. You shouldn't be in a situation where you're repeatedly abused by patrons. "Don't take it personally" -> you shouldn't have to deal with this in the first place.
Regardless, you find yourself here. So—
To echo other replies, in similar situations, I've found it useful to make sure the rules are the target—not myself. The sort of angle that's "I am but a simple clerk, and have no power. It's The Policies :("
Some replies have also mentioned not using "personal" language, and I absolutely second this. People get defensive quickly. Phrase things in terms of "accounts" and "the library", not "you" and "me".
100% agree with everyone else everyone else saying "throw your bosses under the bus"—and I don't mean this in a vindictive way. If these patrons have a problem, they can take it up with management. It's not your job to get treated like a punching bag.
Also, blame the computer, even if that means lying. Tell patrons that computer literally won't allow you to override fines. The safer version is to imply the computer won't let you do it. Because even if you have the ability to override the computer, you can't override the policies. So, effectively, you can't override the fine, and aren't lying.
Just be careful with this one, and don't use it if your management would throw you to the Mercedes-driving lions.
I disagree with whipping out rules about conduct/behavior. That will only escalate a situation; nobody who's already upset will respond well to "please see the rules about treating staff like dirt".
It can be exhausting, all of this is exhausting, but act like you care. Use tone, body language, and some sympathy about whatever asinine situation they believe should exempt them from the rules. Like, everyone involved is aware that these Karens damn well know about late fines. But people respond better when they believe they're being heard.
These techniques work in many customer service jobs.
"It looks like this account is blocked because of fines. I'm so sorry, but the library's policies are [xyz], and I can't circumvent this."
":((( I know, the traffic is just so horrible these days. I really wish something could be done about the congestion on Main Street."
"Sir/Ma'am, I would like to help you, but I genuinely can't override a fine. I don't have the ability."
"If you would like to speak with the [manager title] about this fine, I can get them for you."
One or two people is to be expected. If you are consistently having to deal with this—it's a sign of a systemic problem, imo. The fact that your request to put up a sign was met with an instant "hard no" gives me the impression that management knows there's a problem. Were I in this situation, I would like to know what steps are being taken to fix the problem.
If the library fines, it shouldn't be a surprise. Patrons should be made aware of fines when they open a card. They should be made aware of fines when they check out materials. There should be signage in the library about fines.
However, I do very much know about the demographic you're talking about. Something can be publicized six ways to Sunday, and they'll still pitch a fit when encountering the consequences of their actions. It's always the well-off folk.
Ultimately, libraries shouldn't have late fines. The true solution is to abolish late fines. But, until that point, I hope this helps.