You've been given very good advice but some is just not as good. NEVER be aggressive to your Basenji - it'll have the opposite effect and make her really aggressive. Basenjis never forget! Positive training can be more difficult with an older puppy but perseverance and consistency is the key. In my opinion a Basenji isn't naturally an aggressive dog, this has mainly come about by incorrect handling in the early days by ignorance of the breed. A puppy that leaves it's breeder is often 'spoiled' by the owner who doesn't take much notice of advice given by that breeder on socialisation, correction etc. I can't find evidence that you've consulted the breeder on Lily's current behaviour. Apologies if I am mistaken and I'm not criticising you in any way. You've obviously tried so hard with her. So pleased that you've joined this forum where you'll find advice from many experienced Basenji breeders who give excellent advice and help. I'm sure you'll find that you'll have great success. The other matter of them eating their own poo can mean that she is deficient in some of her nutrirional needs so check this before trying other methods. Eating other animal's poo is quite common and actually, I've never been able to stop this except that some later do. I have sheep with mine and have had little success in stopping my Basenjis eating their poo. I find that some never do it and some keep on until old age! My current remaining dog rarely does this. I wish you the very best for your happy future with Lily. She's lovely.
Personally, I don't believe that being alone is any more difficult than having another puppy. After all, there's always a chance that he doesn't get on with the new one (little or adult). I totally agree that he'll benefit from having more attention and bonding with you. There's plenty of time later to think about having more Basenji company.
I've got the same situation. Now my boy is left alone he no longer appears to be grieving but won't be left on his own now. We are currently only leaving him for 10 minutes or so as long as one of us can hear him and get back if need be. We'll increase the time until we're able to leave him for the necessary period. We intend to use the same routine as when his sister was alive.
I agree with Tanza about the yelping - make it very loud and instant. Don't pull your hand away straight away but you will usually find a noise that really startles him and he'll stop and think. If one particular yelp doesn't help another might. Do it each time he tries. Remember to give him something that he can bite immediately after startling him. He should soon work out that certain things are acceptable.
Have you asked the breeder for help? Having been with others for the first 3 months of his life a Basenji should have had the experience of sibling and particularly his mother's reprimands when he bites them painfully.
Make sure that both your husband and you react in the same way. One of my pups went to a home where the children played very roughly with him and he did become a biter, so I've had that experience.
I am so sorry that you have had all these difficulties and you've certainly been very patient to try to solve his problems without professional help.
However whatever you do, please, please find a Basenji qualified behaviourist. So many who have little knowledge of primitive breeds make the most horrendous mistakes and can, therefore, make matters worse.
I'm assuming you're in the USA where in my opinion you'll be able to find a suitable person. In the UK, it has become increasingly difficult to find a Basenji knowledgeable behaviourist. Although I'm well versed in Basenjis, I'm unwilling to offer any tips about his behaviour as I haven't seen your boy.
I sincerely wish you success in improving his and your lives.
I had a basenji bitch who would never tolerate being shut in a crate. She even got out of a padlocked crate by collapsing it around her and climbing out! Ziggy is obviously smart and my answer would be never to crate him but leave him in a safe room from which anything destructible is already removed.
If he has doggy company even better and as Pawla suggested tire him out. Give him plenty of long interesting walks and leave him with toys etc. I hope that he isn't left alone for too long particularly while he is so young - he may improve when he is older or more accustomed to being left for shorter intervals.
It would be ideal to have somebody you trust to look in on him and take him out for comfort walks.
I don't think she has Basenji in her make-up but she's lovely. I am more au fait with Basenjis than other dogs so couldn't really guess her make-up. Be warned though that as soon as a dog of unknown origins has a curly tail and facial wrinkles people assume Basenji. If you desperately need to know, a DNA test could be the answer but I'm sure you love her whatever (and no wonder!).