Since you are negotiating, I would recommend getting them to file a motion to vacate the judgement in exchange for payment. What they are offering now is to just note that the judgement is satisfied, which won't help your FICO score. You want to get the judgement removed, and to do this it has to be vacated by the court.
The plantiff/CA CAN do this, it would just be some additional work on their part.....
The policy on vacating judgments based on payment of the debt varies with each court.
Some courts will religiously vacate only upon a showing of some legal or procedural deficiency in the judgment itself, while others will entertain an equitable vacate of the judgment based on its satisfaction. I would suggest making a call to the clerk of the judgment court, and have a chat with them about the usual pratice of their court in handling such motions.
If they entertain such motions, it helps if it either comes from the judgment creditor, or they file a statement supporting the motion. That way the court does not have a squealing judgment creditor to be concerned with.
I don't think that disputing would do anything for you, since the judgement won't go away unless it is vacated by the court.
My husband paid of a judgment in 2009 via wage garnishment and we were unaware there was anything else we had to do on our part. I'm not sure if it varies my state or if some law firms automatically notify the courts, but we had to contact the attorney that filed on behalf of the hospital he owed and get a notarized Satisfaction of Judgment and just yesterday I had to take that to the courthouse and pay to have it filed. I am now sending copies of the Satisfaction to the 3 bureaus to get it to report correctly (right now the status just says "Judgment"). Our mortgage LO instructed us to do this and said it should have some bearing on his report.
A judgement is like a collection in that is will ding your FICO score as long as it shows, regardless of whether it is paid/satisified or not.
If you are in mortgage underwriting then you will likely have to take care of any unpaid collections/judgements as a condition of approval. It won't really do anything for your FICO score though.