Tom, I do just that.
The worst thing you could do is tell your bank what you intend to do, just do it.
I have never had a "bank account" as such, always had a building society account with nationwide. I put fewer cheques ( in my name ) in these days, but never had issues, my local branch will except 10 at a time although I often put them in the machine inside the branch. I opened a second current account just for customers internet payments to go into then I transfer them over every few days to my main account by internet banking.
Like you (being a sole trader) I consider these payments my wages and no business of the bank or building society to tell me otherwise.