Size up in vests and / or sleepsuits - the poppers cause compression leaks x
Have you tried different bands of nappies? I find that Tescos tend to leak but lidls are great because they have a more snug fit near the thighs and the rascals nappies are really good x
My boy is normally in size 3, but at night I put him in a size 4 pull up pampers x
Have you tried reusable nappies? You could just do nights if you wanted to. My baby was leaking through disposables all the time so I had to dig mine out and use them again from my first because they are so much better than disposables. I'd recommend seeing if there is a nappy library near you where you could go and I'm sure they'd have recommendations of what is best for a side sleeper.