I don't want to be too much of a wet blanket, but putting over 300 lbs (including the mount) out behind the trailer is asking for it to sway. It's not about balancing the tongue weight, it's about a heavy weight (mass) so far aft acting like a pendulum. You might be fine, and it depends somewhat on the stability of the TV, but mass way out back causes serious stability issues.
Unfortunately, there is no definite yes or no formula, but the general rule is keep the mass centralized and slightly forward of the
axle for good stability. That is different than two heavy weights far apart that sort of balance each other. This is partly true because the rear mass moves much farther left and right than the forward mass, which is much closer to the tow ball. A weight forward and a weight aft don't equal a larger single weight in the middle.
Since there is no good place to carry the bike, you might just have to experiment. If you have a pickup, it could go in the bed. Some carry them on the front, but that is a serious compromise. You could even extend the trailer tongue and carry it on the tongue behind the
propane bottles. A longer trailer tongue is always better and since you weld, that should be considered. Don't know where your batteries are, but the tongue is about the worst place for them. So, if they are on the tongue and in the way, you could move them.
You might consider the
battery moved to somewhere else, the
propane bottles moved to immediately in front of the trailer body, the tongue extended and a bike rack set on the tongue in front of the bottles and behind the jack. Then you'd have good towing stability, easier backing and you could see the bike while driving.