Get a Braintree account with the white label business, ask them to allow you to send custom descriptors (the hard part) and then send the custom descriptor to Braintree's API so what the end user will see in the receipt is the name of the black label store, that way is less obvious the redirection since everything was done in the backend and the user never knows that he/she was charged with another company
I know is possible since I have configured it for some stores (but they did it because they wanted to handle everything with just one account). But this is not bullet proof either, because if your competition ask the bank to start a dispute, Braintree will need to give the data and your account could be closed too.
I think there is no way to have the perfect solution if your competition want you out.
Also, Stripe could have the same functionality but since you said it was locked I don't think you will continue with them.
I know is possible since I have configured it for some stores (but they did it because they wanted to handle everything with just one account). But this is not bullet proof either, because if your competition ask the bank to start a dispute, Braintree will need to give the data and your account could be closed too.
I think there is no way to have the perfect solution if your competition want you out.
Also, Stripe could have the same functionality but since you said it was locked I don't think you will continue with them.