Sounds like you're doing a really good job all things considered. Your experience with the dogs sounds like you're spot on with little Achilles. Dogs are dogs..and Basenjis are extra special dogs 😃 as you can see!
In my experience these dogs NEED & WANT rules & direction. They're happy when someone else calls the shots. My little girls tends to get a little crazy with my male dog so I know what you're going through & while I can't "fix" her mood swings I do try to manage as best we can. When she gets "that look" I'll get her attention or throw a treat at her & she quickly goes into "ooh gotta find the treat" mood…sometimes I'll just pick her up & put her in my lap with her back towards the dog & lots of good praises for being good.
I'm afraid that your daughter & her fiance may not keep up with all the work that you've been doing with Achilles & that could be disasterous for him. He's only 4 months & they really need A LOT of structure at this young age (kinda like kids 🙂 right?) to be well behaved adult dogs. I really hope she follows your lead & what you've been doing. LOTS of pups need to be on a house line for a few months so that they learn the rules & learn how to behave.
Good luck & hang in there...I promise you for all the work you put into this little guy you will get back 100 times over in love & respect.