There is no reason whatsoever that a derailleur cannot be used with a 3 speed hub.

However, the skipping is likely coming from a worn cog or chainring... this is the most common cause of skipping and the wear on the cog is often not visible to the naked eye.

Another possible source of skipping under load is a mal-adjusted or damaged SA hub.

But since you said the drivetrain worked before adding the derailleur, I agree that the most likely cause is worn cog and derailleur giving the ability to skip. New cogs are cheap so that could be the first thing you try.

Next would be eliminating the derailleur and use the horizontal dropouts to tension the chain (don't use the front shifter during this test!) and see if you can make it skip without the derailleur... because that would indicate a problem with the internals of the SA hub.

IT is very unlikely it is wear in the derailleur causing this... you actually don't need teeth on the derailleur jockey wheels to use it as a chain tensioner - some chain tensioners have non-toothed pulleys and they work fine. Also, I doubt going to a single-speed specific chain tensioner will help because most of them are spring loaded just like a derailleur. Some of them are just derailleurs without lateral movement.

edit: ONe more idea: It could be possible that the derailleur is slightly misaligned and is trying to 'shift' the chain to one side or the other, causing it to ride up on top of the cog and slip. If you look at the derailleur from the rear,are the jockey wheels 100% perfectly aligned with the cog? SOmetimes the limit screws are not quite enough to get a derailleur lined up for use as a chain tensioner and you need to rig up a segment of cable to hold the derailleur in place.