Bank would be free to "pursue additional charges" regardless of whether you're in the right state and have the right circumstances going for you. Whether the bank would win is another matter. You haven't provided any relevant details or said where you are, but I will say this -- even if you're in the right place, unless there's a legit ongoing dispute you can point to between you over money owed, this won't work. (If you owe them $10k and you know that's so, you cannot successfully send in a $2k payment marked "payment in full" and expect to get away with stiffing the bank for $8k.)
In a state that recognizes accord and satisfaction, if you'd hired some contractor to do X-Y-Z for $10k and they only did X-Y and you complained about it, with them refusing to do more and instead demanding all the rest of the $10k, and you send them a check for $5k, which is what the X is worth -- noting that it's payment in full either on the check or on a letter accompanying the check, then the company could find itself out of luck if it sued for the balance after cashing the check.
You need to read up about accord and satisfaction as it relates to your unnamed state.