I *think* the train you get will be the same Manchester – Sheffield train that I get from Manchester which goes through Edale, Hope etc.
If this is the case there is no need to pre-book as they are all really old and grubby and often ferry walkers/mtbers etc in and out of the Peak District. There is usually a space for bikes in the end carriage.
It will be busy on a sunny Easter weekend though. That said, I’ve often got on and seen 6,7 or 8 bikes piled up in the bike rack and they’ve been fine with it. They seem pretty understanding and tolerant of the bikes so you’ll be fine, they’re pretty used to it I think. I’ve never had any problems anyway.