X2. If you need to add more weight to the front of the TV, in excess of the unloaded weight, to get rear within specs then you have too much tongue weight or not enough tow vehicle. Something has to give.
Great advice. To the OP there is even more. You need to check your receiver to be certain it can handle the tongue weight. If not you might need to go to a higher rated aftermarket receiver.
A weight distribution hitch will not increase your payload, in fact the weight of the hitch can reducte payload of the TV by 75 -100 pounds or more. Also, a properly tensioned WD hitch, if the rear axle is still over it's rating, then the TV is not up to the towing demands. The hitch can only do so much, it cannot increase payload, axle or gross weight ratings. What a WD hitch does do, is to transfer about 20 percent or so of the tongue weight back to the trailer axles and some back to the front axle. Beyond that it is not magically making something work that is over axle or payload ratings. My last comment- never go by brochure weights or dry weights. If the brochure tongue weight is provided, it likely is at the dry weight. Use gross weights for your calculations. Going to the scales is the right thing for you to do. A 6300 pound trailer should have a tongue weight between 630 and around 900 pounds, not more. 750 pounds would be near ideal. Your hitch will never set right until the tongue weight is within range.