Is the aforementioned method of transferring funds between currency exchange shop owners, which relies on trust and periodic settlement of accounts without a pre-existing balance, Islamically permissible? And what is the legitimate Islamic way to do so?

It is permissible to transfer money for a fee, or to exchange one currency for another with a transfer. If you ask someone to give a sum of money to another person and you do not have an account balance with them, this is considered a loan, and it is not permissible to take an increase on it, because every loan that draws a benefit is usury (riba). The legitimate way is for you to have a balance or to deposit the required amount to be delivered with the other party, and in this case, it is permissible to take a fee because it is an agency and not a loan.