IANAL, and I will gladly yield to a lawyer on this one, but what you are proposing sounds to me like pretty much the definition of forging a check. I really doubt that "but they owe me the money" will hold up in court. Frankly I think the judge would put that in the same category as someone who robs a store and says "but I was just getting back what they stole from me when they overcharged me last month".
I was taught in school many years ago that if a check is made out to you with an incorrect name -- a mis-spelling or a nickname or a maiden name or whatever -- to endorse it with the incorrect name, and then endorse it with the correct name. I've only done this once that I recall. The check was made out to me with my correct last name but using a nickname rather than my real first name. The cashier questioned it and took it to the manager, but the manager then approved it. The check was for $9,000, and the cashier told me that if it hadn't been so large it would have been a non-issue. Ah, here, this web page says the same thing: https://pocketsense.com/can-cash-check-name-spelled-wrong-8970.html (The text of the article mentions issues beyond a simple mis-spelling.)
If your bank refuses to accept such a check, you have two choices: Get the originator to send you a corrected check (which apparently you have tried to do and have been unsuccessful), or get another bank. I don't know what else you could do. You could dig out the relevant laws and show them that this is legal, but I don't think there's any law REQUIRING a bank to accept any check. Even if there is, you'd likely get bogged down arguing legal technicalities with the bank manager.