Kinda shooting blindfolded not knowing which side of the bale the knot is breaking, so here's a laundry list of "possibles" from my experience..... 

1. see if the hay dogs are worn more on one side and check the hay dog springs,

2. flip up the knotters and very closely inspect the bill hooks. (I looked at the bill hooks on the Freeman 330 and they "looked" ok,, but after driving out the offending billhook's roll pin,, I found the dang thing was broke.) I replaced the roll pin and the bill hook worked perfectly again. Looked fine physically but under the stress of tying a knot, the hook wouldn't hold the twine properly..

3. check the twine disk holders, the offending side may not be holding the twine tight enough when the billhook is tying a knot. Tighten the twine holder springs. You may have to advance or retard the twine holder, check to see they are set where they should be.

4. Check the twine knives,, they should be sharp enough to shave with.

5. Check the arm the knife is on, it should just "barely" clear the bill hook to sweep the knot off the bill hook. (with a freeman "they" suggest the arm should actually just "tick" the bottom of the bill hook as it passes by sweeping the completed knot off the billhook) If memory serves,, the "adjustment" to the arm is done with a big Crescent wrench and arm power so the knife arm is close enough to the billhook. Check the bearings on the ends of the arms,they should roll "silky smooth"

6. Check the knotter gears on the billhook, etc.. make sure they are worn equally. (I found the bill hook gear worn more on 1 than the other two which was just enough to allow the bill hook to turn just enough slower to miss the twine, and miss the tie.)

6. Get a bale out of the baler before it goes through the thrower,, or watch so you know exactly which way the bale came out of the baler,, and see if the tension is the same on both strings.  The feed forks may be packing more hay on the right hand side (as you're facing forward on the tractor seat) or on the left hand side.  If the right side string is tighter, move the bale chamber feed forks to the left.If the Left side of the bale is tighter, move the feed forks to the right.  (The feed forks for the bale chamber move the opposite of what seems logical to adjust where the hay is placed in the bale chamber. it's about how much hay is forked into the bale chamber. Feed forks moved "right" more hay on left side of bale, move feed forks left, more hay on right side of bale.)

A 50# bale really isn't all that much weight, etc.. 240 lb twine should have adaquate knot strength to pack lightweight bales.

As stated,, I've never seen a bale thrower work, shoot only seen 1-2 pictures of them but I seriously doubt the bale thrower is breaking the bales.

On second reading of your post,, first try moving the first set of feeder forks (the ones that go fartherest into the bale chamber) to the right a few inches. That will place more hay on the left side of the bale.

I've had something similar happen to me with twine breaking very close to the knot. It turned out to be a burr on the knot wiper arm. Upon close examination of the string, about 30% of the string fibers had been cut, no more than about 3/8" behind the knot.

The problem turned out to be the knot wiper arm. On the Hesston baler, these are "adjusted" with a hammer - you whack them until they're just contacting the billhook. As the arm would come across, it rubs on the billhook, and over time, this creates a slight burr on the edge of the wiper arm.

We simply used a round file and took the burr down.