Eric, that's bad. You do not want that. It will show a negative on your paypal experience and they will show you as someone that's had a questionable transaction, which, in the future, can sway their decisions in future issues.
Take the proof of the new payment and submit it to Paypal as supporting documents for the case. You do not want this to go against you.
I know mathematically, it all works out the same, but basically someone has told Paypal you do shotty work, and now that's "on your record". You need that off your record. In fact, if you resolve it in your favor, through their process, it can make your cases stronger in the future.
Why would that person make another payment instead of retracting their claim? Doesn't make sense. All they need to do is go through the same dispute resolution center and tell them the case has been resolved and they are fine with it.
Something's not right. Make sure it's resolved THROUGH paypal, not through additional payments.