Ruling A very good stall deck strategy if this is allowed...

[LucarioXFroslass]

If the defending pokemon is asleep, can I use Unown E to keep them asleep?

Hidden Power (PSY/COL) During your opponent's next turn, whenever your opponent flips a coin, treat it as tails.

 

[PMJ]

Sleep checks happen between turns. Unown E says "During your opponent's next turn..."

Unown E's effect doesn't take effect until the between turns phase is over (when sleep check happens) and your opponent's turn actually begins.

So no, this won't work.