The fact that a grey water discharge line is addressed in the National Plumbing Code does not mean that it is legal in any particular county. Each county adopts its own set of building and septic codes, and to violate them is illegal. Most counties in and around urban and suburban areas have specifically outlawed grey water lines.
That having been said, the solid pipe away from the house connected to a French drain is definitely the answer. And no, you don't need a backhoe. I am 57 and have dug two long french drains by hand in the past 3 years. Since this ground is already wet, it would not be too difficult.
The secret to digging a good trench is that you don't shovel down to the final depth all at once. You take a medium size mattox, with a hoe on one side, and straddle the path of the ditch. Then, you lightly swing the hoe down into the soil, and lift up the handle. When you do this, 2 inches of soil will be lifted upward by the hoe. Move over and do about four of these cuts in a side by side row. This will give you a one foot wide ditch.
Now, step backwards about 6 inches, and repeat the process. After you make the 4 cuts and lifts, step back again and do it again. Just keep moving backwards until you have done this over then entire length of the ditch.
NOW, get out your shovel, and going foreward, shovel out the loose soil and lay it neatly on one side of the ditch. Don't throw it, just lift and place it. You will find that it is pretty easy to take out the top 3 inches of dirt for the entire length of the ditch.
Now, start the entire process again, using the mattox to chop and lift the soil, moving backwards, and the shovel to lift it out, moving forewards.
Set a pace, even a slow one, that lets you keep working like a machine. If you need to work an hour, and then rest an hour, that is totally fine.
After you have done this a total of 4 times, you will have a one foot wide and one foot deep ditch.
If you can find one good helper (a strong teenage boy), it goes pretty fast, because you can trade off using the two tools. And while one person is chopping and lifting the soil with the mattox, the other, traveling towards him, can be lifting the loose soil out with the shovel.
The keypoint is, it ain't a race. If it takes one day, then fine. If it takes three days, then fine. You will be very proud when you are done.
P.S.- I learned this system by watching two old Mexican men dig a long ditch one day. They got into a pace, and never stopped working. It was pretty amazing. At the end of the day, I offered them each a cigar and a couple of cold beers. They refused, saying "No patron, no." I guess that in Mexico, you just don't socialize with the boss.
Best, Mannyrock