Banks specifically DON'T want you doing that because they don't want undesirable businesses funneling money in and out of their banks
Not sure I understand this objection - the recipient of such a transfer would already have to have an account at another bank, and that bank would have to do the full AML/KYC checks anyway. There should be no issues with someone with an account at reputable bank A sending a regular payment to an account at reputable bank B for any purpose.