You have a brake problem such as out of round drums. If the pulsing is worse at low speed and not as noticeable at high speed the drums may not be concentric or out of round. Pull the drums and check for concentricity and or for grease on brake shoes.
I had a new trailer with Lippert axles that had two drums that were not concentric by about 1/8 inch but were round. They were not machined correctly when manufactured.
If it is not the brakes you have a defective controller. A prodigy should not do what you describe regardless of the gain setting. Give Tekonsha a call as they are very helpful with controller problems. I am sure they will agree a Prodigy that is not defective will not do what you describe. If the controller is defective they will replace it at no cost to you.