There are a couple of compelling reasons that I think you should go ahead and pay.
One is that even though your score won't increase, a "paid" judgment looks much better on a manual review (like for a mortgage) than "unpaid". I've seen anecdotes on this site where someone is required to pay off a judgment before the mortgage closes.
Another reason to pay is that some states allow an unpaid judgment to be renewed after a certain number of years. Which means that you could start the clock again on a ~new~ judgment. Check your state resources.
Also, and maybe the mods can comment on this, but my understanding of how the reporting works is that judgments are unlike CA's in that they don't keep updating. So where paying a CA, or forcing them to correct some incorrect reporting, can make it look like a more recent collection in terms of your score, judgments don't work that way. Judgments are not reported as 90-days late or whatever each month, or have a DOLA, or anything like that--I don't think there's a mechanism for doing so. So paying off a judgment wouldn't cause the judgment to update as a newer judgment (like CA's can make it look like a newer collection) except that it would be marked "paid". Admittedly, "paid" doesn't really help you in terms of score.
Another thing is that it is the judgment itself that hurts you--it doesn't matter how quick you pay it off, nothing about you pay it--just "paid" "settled" or "settled for less than full amount" and the date it was paid (or the "paid" document was filed with the court). So I don't see how paying it off can negatively affect you because of reporting newer. Also, where you can't vacate, the fall off date counts from the date of the judgment. So even if paying it somehow makes it look like a newer judgment (which I don't think it does), it still falls when it is supposed to. Paying definitely does not make it stay on longer. Mods?
Your judgment is for a relatively small amount, so it seems unlikely that they'll garnish you or freeze your accounts or use any of their harsher remedies, but it might be worth the piece of mind.
