The fourth level of exhaustion imparts this effect on a creature:
Hit point maximum halved
The Basic Rules make it clear that whenever you divide a number for any purpose, you round down:
There’s one more general rule you need to know at the outset. Whenever you divide a number in the game, round down if you end up with a fraction, even if the fraction is one-half or greater.
Most Challenge rating 0 animals have a single hit point, so when these creatures reach the 4th level of exhaustion, they would presumably be left with 0 hit points and die before reaching rank 6. Is this correct?