The tongue weight is needed for the sway you mentioned after loading the bikes. You may have severely unloaded the tongue.
How to get weight forward? Add a second battery, pack personal items toward the front of the trailer. For test purposes you can use most anything to get an idea how the trailer will settle.
As to the bounce, the next suggestion would be to look at shocks on the trailer to slow the rebounding. You could also soften the tires slightly since you said you were at the max.
Looking at the pics of the unit, it certainly looks like the axles are far enough back to avoid lightening the nose too much, but I would still try to get some weights. Also hopefully your water tank is forward to aid in loading.