You've done well troubleshooting. You're applying a load perpendicular to the axle (which it should handle) and causing a screech. Does the pitch rise with speed? To me it sounds an awful lot like you have contact between the hub and something on the axle.
Since you recently replaced the balls in your hub, I'm going to guess your cones are out of adjustment and you have too much play between your axle and the hub. Well adjusted bearings should be able to resist this force since it's the force directions they're designed for. Hopefully, it's just a dust cap making contact with the exterior of the hub. I would expect the sound to decrease after use, as you grind away the surfaces contacting each other. A bent axle will also make it difficult to adjust the cones, and would move them out of alignment.
Either way, it seems like you'll have to take a good look at your hub bearings, check for play in the axle, and possibly pull the axle to roll on a flat surface to check if it's bent.