Sounds like you've got two issues going on, not pooping on lead and only pooping in one specific area.
I think the first thing to work on is pooping on lead. Some dogs can feel pressured by you being so close if they are not used to it, so using a longer lead (10 - 15 feet) might be a good idea. Do you have a potty cue (mine is "go hurry')? If not, then start using one. Take him to his preferred spot, and as soon as you see him starting to act like he's going to poop, give your cue. Once he's done, praise and treat like you would have when you were first house training him. When he looks like he's made the connection between the cue and going, try it with the long lead. If he doesn't go after a few cues, take him back inside, and try again in 15 to 30 minutes.
When he is consistently going on lead in his preferred location, start taking him to other places, again with the longer lead, give your cue, and wait to see if he goes. If he does, praise and treat. If he doesn't, take him back inside, wait 15 to 30 minutes, and take him out again, go to the new spot, and repeat.
Once he can potty in different places on the longer lead, you can try a shorter lead.
In my opinion, teaching them that they can potty on lead in different places and on different surfaces is a good idea. There is nothing quite like taking a city dog, used to pottying on grass, through the desert, and her being so appalled at having to pee on bare, sandy dirt, that she doesn't pee for over six hours.