Success! I've got the quilt together and the centres of top and back are perfectly aligned. I put my largest cutting mat in the middle of the floor, because I was working on a carpet, and taped a button on it at the centre point. I then measured and marked the centres of each side of back, batting and top with chalk so that I could line everything up, and placed the backing with the centre over the button. I taped the back to the floor with the tape in line with the markings, so that could be an extra check. It helped that I'd added several extra inches to the size of the back, knowing that sandwiching was going to be tricky. I could then cut the batting slightly smaller than the back, which made it easier to see and line up my marked centres.
Thanks to everyone who gave me their tips and advice. It was a big help, and I'm hoping that, pieced back or not, I won't dread putting my quilts together in future.