If it was me, what I would do on a trailer that short and that weight:
Hitch the trailer up and then stand off to the side and look at it in relation to your tow vehicle.
If it is perfectly balanced so that it is not down at the tongue or up at the tongue and your tow vehicle is not down where it is connected, I would pass on the weight distribution bars.
Then I would test tow it at various speeds. If it wants to sway, I would get a sway bar. If it tows straight behind you at all speeds I would pass on one.
Keep in mind that when you are towing at higher speeds and a eighteen wheeler passes you, you are liable to get some sway. If it is a lot, then get a sway bar.
Good luck
Jack L