I think basenjis are really "puppies" until about 18 months. It's funny, because they actually look like little adults at 6 months, and yet at 15 years old, people would ask how old my puppy was (I think they age well).
Please don't give up on Lilly - this too shall pass (eventually...).
Accentuate the positive, don't focus on the bad. If you can, find a way to take a break once in awhile.
As previously mentioned, there is something you can put in the food to make them stop eating poop. Always be the gentle but firm leader. They know the difference between fair and unfair. Going for a long walk before doing anything that requires good behavior is excellent advice - before leaving him/her alone, before an obedience class or going to the vet, etc. Taking an obedience class (look for a local obedience club?) is good training for YOU - not necessarily the b. You might get some ideas on how to work with a dog, but the main idea is to be consistent in the way you make your request. My b is extremely stubborn and often will "pull a mule" if he doesn't want to go in the direction I want - to the point that he will back out of whatever collar or harness he is wearing. My eyes are always on him and I never let him walk behind me. (Ceasar Millan would not approve :->.) Sometimes, I pick him up and carry him a bit, sometimes we just cross the street (and he thinks he won?) I tell people these dogs are not for everyone, they are more primitive than other breeds, and they don't take to training very well, but when they love you, you KNOW it is their idea and not just because you feed them. Keep us posted on how it is going with your little one - what works and what doesn't - it could be helpful to us all.
My condolences to those who recently lost furkids.
I am guessing that KhaiPan would not be interested in a puppy because of being at work all day. Doggie daycare is a good idea, but costly (also, you have to be careful it is a good fit for Max) . Another thought would be to consider adopting an older (tolerant) adult dog (of any breed) from a shelter or rescue group to keep Max company.
I hope he's okay. So sorry for your loss.
Go slowly. It took me 8 months to get my adopted adult basenji and older cat to coexist. They still have their moments, but I can trust there will be no bloodshed. You are lucky that you already know your b's accepted the bird. That is a great sign that you will be successful.