A butler can use an ordinary bed. They can certainly use a double bed, too. Whether they'll share that bed with another, non-friend sim, or whether they can use a bunk bed, I'm not sure. I usually give him or her the cheapest one outdoors where they won't bother me.
![Cheesy :D](https://www.carls-sims-4-guide.com/forum/Smileys/default/cheesy.gif)
Edited: I found this response in regards to butlers using beds on the Sims 3 forum.
Yes, my sim's butler obviously likes the bunk-beds in their house because he climbs up there and goes to sleep on the top and they have to wake him up because that's someones bed. Silly butler!
It doesn't answer whether you can assign the upper bunk to the butler or not, but it's obviously possible for them to use it on free will.