Multiple questions here. Why don’t you have enough staff that if one person can’t be there you’d still be fine? Why does this person feel the need to get another job in the first place? Could you hire more people or give this person more hours?

Since this is a part time job for this person they very well may decide it’s not worth the hassle and just quit anyway if you decide to say no. You have every right to say no to the schedule change but your employee has every right to quit as well. You aren’t a bad person for having your reasons for wanting to say no. But understand that your employees don’t own this company and they don’t have the same stakes in it that you do.