I've used a lot of the white powder floor leveler over the years in old houses. Mix some with water. put it down with a trowel. It sets up fast.
Around toilets and showers, if at all possible, I coat everything with epoxy before putting the floor down. I'd never use particle board anywhere near water. That just is a curse.
In a rental property, I wouldn't be worrying about 1/32nd. I'd have a hard time getting exercised over 1/8th.

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1977 300d 70k--sold 08
1985 300TD 185k+
1984 307d 126k--sold 8/03
1985 409d 65k--sold 06
1984 300SD 315k--daughter's car
1979 300SD 122k--sold 2/11
1999 Fuso FG Expedition Camper
1993 GMC Sierra 6.5 TD 4x4
1982 Bluebird Wanderlodge CAT 3208--Sold 2/13