A question regarding dog training

We’ve had a few dogs over the years and housebreaking them was never a big problem. But now we got my mother-in-law’s puppy labradoodle – who is a real sweetheart – for a couple of weeks to see if we can housebreak him because she was not successful.
My wife turned out to be a better animal psychologist than I am and figured out what the problem is. This is not a case of a little puppy who is not yet housebroken. This is a case of a puppy that was inadvertenly trained to poop inside the house and not outside. What we think happened is either that the dog got yelled at when he soiled the carpet, or something unfortunately scary happened when he was doing it outdoors. He seems to be worse about it now thahn he was two months ago. Now, he appears to be afraid of doing it when a human is watching. And out on a leash-walk, a human is always watching. Indoors, there can always be a moment when one can sneak away and do it in the bathroom (or wherever the door is not closed at the time).
He’s been with us for a week and he managed to poop outside only once – for my wife – and promptly got rewarded and fussed over. But once is not enough. Only once I managed to catch him in the act and, without any anger, I quickly took him for a long walk during which he did NOT do it!
Now, if I had a fenced in yard, I’d let him out – it’s been in the 80s here lately so it is warm even during the night – and keep an eye on him through the window so, if I saw him pooping outside, I could get out quickly to give him a big reward. But I do not have a yard at all – we live in an appartment complex, so even tying him up on a very long rope is out of the question. Perhaps getting one of those super-long leashes would do the trick as he could get 20, 30 or 40 yards away from me (forgetting the whole training on how to walk properly on a leash) and do his tricks without me hovering over him – perhaps one of those fancy leashes with the button to haul him back for a reward afterwards.
I will have him for another week. After that he has to go back so we can thoroughly clean the house and wash the carpets in time for the Passover dinner (perhaps he can come back for a second course afterwards). And I am out of ideas.
I don’t know if Christie still reads my blog, but I assume that some of my readers have some experience and knowledge and good ideas about what to do. Shoot them in the comments.