What is the ruling on delaying entry to the restroom until after the prayer due to the sensation of drops of urine exiting during it, knowing that relieving oneself before the prayer would lead to delaying the prayer until sunrise?

If delaying entry to the restroom until after the prayer does not cause you hardship, then it is commendable for you to delay it. However, if it does cause you hardship, your ruling is that of one with an excuse, so you pray in your state even if something is emitted from you. Sheikh al-Islam Ibn Taymiyyah was asked about a man from whom wind is emitted only during prayer. He replied that his ruling is that of people with excuses: he performs ablution and prays, and what is emitted from him during prayer does not harm him. The most he needs to do is perform ablution for each prayer.