Need more detail really but it doesnt sound much like a short. It sounds more likley to be a high resistance connection. If it all looks good at the dash it could be detecting a resistance and can't tell this apart from a 'load'. All the dash knows is that it's sending 12v down the wire and it is being 'dropped' across something but it cant tell the difference between a high resistance connection and the load (Light bulb) intended.
An open circuit would not indicate properly at the dash and a short circuit SHOULD rupture the fuse.
I would initially check the trailer earth circuit as this could be dropping some of the voltage and leaving very little for the lights. The trailer is usually earthed through the tow vehicle so relies upon a good earth there.
Have you got a trailer board that you could plug in to check?
Have you checked the socket pins/sockets for cleanliness and tightness?
Have you got a mulitmeter to check continuity and insulation in the trailer loom? If for example the trailer brake light cable is chaffed through the tail light cable, when the tail lights are on the power will also be shared with the brake lights and they might be so dim that you cant spot it but they will still use the power.
Remember KISS (Keep it simple stupid) and look for easy wins. The fuses, earths and connections would be the simplest to check first along with bulb holder cleanliness, correct wattage filaments etc.
Oh! And good luck!
