Perhaps Snape is made to attend some sort of counseling or mentoring sessions. Dumbledore might think that there is some underlying issue that is causing Snape to bully Harry. So, he arranges for Snape to talk to someone about his feelings and actions. This would be a way to reform Snape rather than just punishing him. And if Snape refuses to comply with any of these consequences, Dumbledore might have to take more severe actions like restricting his access to certain areas of the school or even considering his dismissal from Hogwarts, although the latter would be a very extreme measure.
In the fanfic, Snape could be assigned extra duties as a punishment for bullying Harry. For example, he might have to oversee detention for a longer period or help with some menial tasks around the school that he wouldn't normally do. This would not only be a punishment but also a way to keep him in check and away from Harry. Also, his reputation among the students might take a hit. If the students find out that Dumbledore has stopped Snape from bullying Harry, they might start to view Snape in a more negative light, which could be a consequence for his pride as well.
Snape might be given a warning by Dumbledore at first. Just a simple 'stop this behavior or else' kind of warning.