I think you're showing exactly how much you DO know by asking good questions of the right people here on the forum.
You obviously care about this little horse and that is all to your credit.
I know you have the lady's permission, but Freeland is right: you are not going to be able to tell the daughter to stay in the house or be calling her lazy, so becareful about that too.
Freeland and Marty both missed the fact that all you were saying was that you were not inclined to invite her
out to participate, not that you were planning on ordering her away from her own horse.
I'm with you- if this girl has no interest, it isn't going to do anyone any good to try and force her to do something. The mom is happy you are working with the pony and that's all that counts.
I also respectfully disagree that once a week is not enough to be worth bothering with as far as training goes. No, you aren't going to be able to really condition him in that time. But any exercise is better than none and many students around the country learn plenty in weekly horseback lessons. It has been proven that horses need far less repetition then previously thought to learn something; that is part of the prey mentality. Learn it the first time or you're someone's lunch!
So on your actual issues, here is what I'd recommend:
1.) He is afraid of fly spray.
2.) Building the trust level
3.) Need to buy a cheap lunge line (where at and a link?)
4.) Need a workout schedule to trim off his winter fat.
The clothes line idea sounds fine provided you have some heavy gloves to wear to cushion your hands. That stuff will give you rope burn like crazy! Heavier nylon rope could probably be purchased at your local hardware store and a swivel snap attached to it too. On the other hand, for all I know lunge lines can be bought for $5.
Both of mine my mother had before I was born so I've never needed to buy one.
Workout schedule- not gonna happen I'm afraid. Lunge him for a few minutes each way at a walk and trot until you see what his current fitness level is, then take it from there. Don't stress him, just work on voice commands and getting him out there enjoying himself.
Build the trust level BEFORE tackling the fly spray issue. Use wipes in the meantime. Just spending lots of time grooming this horse will start that process. Hang around with him, give him treats, feed him his dinner if the owners will let you. The way to a horse's heart is ever through his stomach.
Having you DO things with him when no one else is will help. And every time you praise him for doing something right he will come to trust you a little bit more.
When you have him looking to you for leadership and enjoying your company, work on the fly spray. I'd start with a small squirt bottle full of water and do whatever seems appropriate at the time. Spray him with the cool water on a hot day, spray yourself while standing near him, spray the air around him until he quits flinching at the sound. The first time you go to spray HIM, stand close by his shoulder with a hand on his neck or withers. Start squirting the air near him and then nonchalantly move the bottle towards him if he is calm. The minute he stands quietly (no matter how nervous!) and lets the mist settle on his skin, tell him he's a good boy and stroke him firmly! It seems to counteract the tickle.
Don't press for more, just stop and love all over him. Do something else like groom him for a minute, then try it on the other side. He may get it quickly from that point or you may have to stop there and repeat the next day before moving to more sensitive areas. Let the horse tell you.
I have also been known once they are okay with the spraying sound to bend down and spray something on ground near their feet (like a particular pebble), paying no attention to the horse. It's the same way you get near stray cats- they won't let you within a mile if they know you're coming for THEM, but I can walk within a foot of that same cat if I ignore them and make it clear they're merely a piece of scenery on my planned route. The horse probably isn't so much worried about the actual fly spray as he is the fact that you are pointing it at him and then squeezing the trigger. Plus the mist feels funny, of course. Make it "part of his scenery" and he'll soon quit worrying about it so much.
Hope this helps!
Leia