I can't imagine having dogs and not having a fenced yard. Do your landlords understand that a fenced yard would be a significant improvement to their property? Maybe you could look into that and present it to them in that light. I can't believe the neighbor is volunteering to build you a fence...fences are expensive. What kind of fence does he have in mind?
When I was renting my house before I bought it, I rented temporary construction fence panels and used it to fence the yard. That's also a possibility.
When reading through all this, I'm sorry, but it was your daughter-in-law and the five-year-old that irritated me, not the old dude. I have a neighbor who had a similar thing, her daughter-in-law was at her house and the bratty granddaughter opened the door, but it wasn't a dog that got out, it was a bird. I'll never forget my poor neighbor, walking all around the neighborhood, crying, and calling out, "Peepers! Peepers!" She said to me, "Why can't people control their kids!!!!" And it was her own grandkid, but she wasn't afraid to put the blame where it really belonged.
I have five dogs, including two Dobermans, and I damn well make sure they never get loose. In my neighborhood, people have shot dogs who were running loose and perceived to be a threat. It's made clear to any visitors at my house that those two dobies are the center of my universe and woe unto anyone who would endanger them.
