As others have mentioned, the decision to remove you from ChexSystems is up to the bank that put you on there.
I suggest that you speak to a manager at that bank’s branch and find out what they are willing and/or able to do. If you can pay them back, and the amount isn’t large, they may be willing to remove you. As an assistant branch manager, I was allowed to make that decision. I wanted to know why the account was reported. For example, the amount of the balance owed, whether it was the only time the account had gone negative, and while this last one may not be the best reason, if the person was polite when they asked for help, I would look upon their request more favorably. I’ve never been find of rude, pushy, overbearing people! Be prepared for them to not do anything as they may have a policy that they refuse anyone unless it’s a bank error.
As for what you do now, I would suggest that you ask your current bank if you can open a savings account with no ATM card access. It would be a pain, but if you can keep their required balance for check cashing all banks are different then you would at least have an account. If they say no, then you will either need to sign checks over to your partner or see if you can cash them at the grocery store. (I don’t know what the fee might be.) Or, you can probably go to the bank your check is drawn on and cash it for a fee. You will probably have to show two forms of ID, and maybe put your thumbprint on the check. Typically the fee at the bank will be less than the fee charged at a check cashing business.