Just to further confuse the issue, although it may shed some light on OP's problem actually, I had a disappearing fine for three days recently until I sorted it out by a blatantly meta method.
Fine was gained by abandoning a mission; I had some cargo to haul, saw another cargo mission which was offering some engineer stuff I needed right then but didn't have enough remaining cargo space to take it, weighed up the pros and cons and abandonedthe first mission before dumping their cargo in a station. As expected, 18k fine for mission failure from the original mission-giving faction with a 7 day timer on it.
I did the second cargo mission, got my engineer stuff, headed off to pick up my Vulture and onwards to my old mate the Dweller, Vulture upgraded, logged off.
Logged on the following day and because I'm so damn nice and always pay my fines, first order of business was to head to the system where the faction have a presence where I could pay it off.
Checked my transactions tab just to be sure and yeah, there it was - 18k fine, 6 days and some hours.
Plotted course, jumped to system, landed at station, went to contacts in the station and had no option to pay the fine.
Checked my transactions tab and the fine isn't there any more. Not one to worry about things like this too much, I headed off to do other things.
Day after I logged in and set off to do some deliveries for the CG. Picked up, arrived, dropped off and checked my transactions tab to make sure the delivery had been credited, what the actual frak??? My 18k fine is back. Note this wasn't from the CG faction or anything, the fine is from a minor faction in a system about 200 LY away that I hadn't visited at all that day. The timer is now 5 days and some hours, obviously the same original fine.
Having played this game for nearly a year I'm nothing like daft enough to immediately jump back to the fine system again. I worked out that the last time I'd definitely seen it in my tab the day before was about 30 minutes after I'd logged on and I'd been playing for about 15 minutes at this time, so I did another few trade runs. Checked my tab after about 45 minutes and sure enough the fine has vanished again. Carried on playing for the night, logged off.
Logged back on yesterday, checked transactions and imagine my complete lack of surprise when I saw the fine there again now with 4 days left to run. That system is actually one that I want to go back to anyway, so I decided to fly over. It was quite a trip. On arrival checked my tab and yes, fine has vanished and contacts at the station doesn't give me an option to pay it.
So I logged out. Logged back in and there it was in my tab. Hit contacts and there's the option to pay it, so I did. All of which shows that even taking account of the convoluted routes which fines and bounties are supposed to follow, there definitely seemed to be something buggy about that partcular fine.