NO, do not push him.
Night time dryness, is a physiological, development.
And it occurs, up until 7 years old. To develop that physiological maturity of the organs.
You cannot train, for night time.
It is completely different, than daytime pottying.
Put it this way: ALL of my daughter's Teachers, from preschool to Kindergarten to 1st grade, ALL said, that kids these ages wear diapers at night, it is normal, and it has to do with their body's development. AND kids these ages, DO have pee accidents, day or night... and even at school. And this is normal. Teachers do not flinch about it. Nor make the child feel bad about it.
Even the health room at my kids' school... they have TONS of extra underwear and clothing, for kids to change into, if they pee or have an accident.
Of which, you cannot change or speed up, physiological development.
Just get a few waterproof bed pads, to put under him at night.
I got mine from Amazon. I have 4 of them and just rotate them as needed. The sheets do not get soiled that way.
My son is 5, and still wet at night. He wears Huggies night time diapers.
AND I sometimes have to change it, during the night. Because it gets soaked.
My daughter, even at 7 years old, had pee accidents. Normal.
Pull-ups are useless at night... because it leaks and is not meant to hold a lot of pee. It is for daytime. Unless you get the nighttime version.
When my kids ask, when they will wear underwear at night... I simply explained to them, that their BODY... is not ready yet. And it is okay and normal. When their body grows up, then they will know and then only underwear can be worn. It has never been a problem with my kids.
My daughter, even had a dream that she was on the toilet... until she felt herself wet. She had peed in her sleep. She didn't do it on purpose. We laughed about it. Changed her jammies. Changed the bedpad, and she went back to sleep.