This is the problem with using the "big kids don't wear diapers" comments when potty training a kid. They sense our desire to have them grow up, and they feel the sense of pride but also our sense of urgency. We are so sick and tired of diapers, and when our kids show interest in, and competence with, the toilet, we regale them with praise.
But we aren't so good at communicating to them in 3-year-old terms, the concept that this just isn't about training! It's totally developmental. Feeling the urge when awake is totally different from waking up to pee and from having enough bladder capacity to go all night. We really don't want our kids to get up several times to pee - we want them the sleep! They need their rest.
Your son is already caught up in the "big boys don't wear diapers" or "pull-ups" so I don't know if you can convince him that there are different rules or goals for nighttime vs. daytime. Nobody's going to sleep well if there are wet sheets - never mind the nightmare of laundry the next day!
If you can work some compromise with pull-ups for night, great. Otherwise, the rubber pants will save the sheets but not keep him comfortable because he will stay drenched. There are pads you can get to put in night-time undies (for those kids who wet through them) so maybe he will allow that to be slipped into his underpants, or maybe you can do it after he falls asleep if he's a sound sleeper.
We put a fitted sheet on our son's bed, then a waterproof pad, then a second fitted sheet. It at least left a clean, dry sheet on the bed we could use after we stripped off the wet stuff and wiped him down, but it wasn't ideal.
So I'd show him the pictures of the big boys on the packages of Good Nights or whatever brand you use, and tell him this is specially for big boys like him, as an in-between stage. Maybe there are photos on their websites too.