Pecking orders change, it's spring...longer days might be getting the hormones riled up.
As long as the picked on bird is not bleeding or chased, pinned down and abused, I'd leave it all be.
Chickens peck lower status members of the flock all the time every day, it's just how they roll.
As long as they can get away when pecked they just go back to their normal activity, part of which is being pecked regularly by higher status birds.
One thing to think about is maybe the egglu is too small (all the ones I've seen are pretty tight quarters),
Chickens do not like to be crowded, might be why the ruckus starts when the low bird goes into lay.
The coop should be big enough for everyone to have plenty of room if they have to be confined to the coop because of predators, bad weather or any other reason....like you not being able to open the coop door the minute they wake up and start to go about their day.
Another thing that sticks out is that you said you feed them morning and night?
They should have a good balanced chicken feed available all the time. That can help pecking order problems as everyone takes their turn at the feeder in order of flock status. If you feed only at certain times, is creates a rush to the food and the competitive survival part of the flock dynamics rears it's ugly head. Multiple feed and water stations is one of the best tools when integrating new birds because of this urge to protect their resources for survival-food and water and a place to lay.
Splitting the run and building another coop(with adequate shelter and a nest to lay in) is a good solution to try....
...and even if it doesn't solve this problem you will have a separate enclosure to integrate new birds if it comes to that. If you have to get rid of birds, I'd get rid of the mean ones.