£3 flat sounds basically impossible become profitable on. Especially considering you're going for higher volume customers. Yes you have chosen a segment with low overheads (bank to bank transfers), but you will still have the burden of costs that scale very poorly with the volume of £ moved and number of customers served (e.g. CS, AML, KYC etc). Usually revenue (and some costs) that scales with volume is the route to profitability here, but you've explicitly forgone it!
So, in my opinion, your prices will have to rise substantially (most likely) or you will do something akin to what Robinhood did with PFOF and make the customer the product, which is to say, exploit them without them knowing. Sorry if this sounds harsh but your setup currently falls into the category of things that sound too good to be true, would love to hear how you plan to overcome it.
Having said all this I wish you the best of luck, hope to be proven wrong!
(edit: to clarify, i'm considering a business model without significant cross subsidisation - in which case this transfer product would still be loss making in isolation)