I suggest you Google "Moneysoft BACS" to get up to speed with the various possibilities. That should return half a dozen helpful answers from this very site, and perhaps give a few pointers to other issues to consider. And if it doesn't, you'll have to put in considerable extra effort and Google "site:accountingweb.co.uk Moneysoft BACS" which will give you 84 hits ... and counting. Although it did take nearly 0.5 seconds!
In a case with which I am very familiar, the payroll service provider calculates the sum due to HMRC and submits the RTI reports, but the client actually pays. It seems to be more usual for the service provider also to transmit the payroll via BACS to the clients bank, having the necessary signatory rights on the client's bank account.
In other words, either scenario applies, you choose.