Who is the breeder? Nationals are starting this week so some people have already started their trip. It is possible that is why you have not received a response.
Her name is Nichole. I don't think she regularly attends events but I've seen one record of Logan's dad (Indiana Bones) participating in an event.
How is Logan's relationship with your father and brother? Will he be happy to be "their" dog? I think leaving him outside with a companion could work, but in your position I would see whether another suitable home placement is feasible unless either your father or brother really likes the dog well enough to want to put in the time necessary with a Basenji, i.e. do they really want to have a dog, or in this case two dogs to take care of for the foreseeable future?
He's happy with my dad and brother. My brother watches him the most when everyone's out working. I do the hygiene, training, pottying, cleaning, feeding, and exercise; my dad and brother sparingly help with pottying and walking. He lets out a few deep gurgles when either I or my dad comes home from work. My mom likes him when he's behaved; not so much when she sees his little furs around the house or chewed up door moldings and shoes. She's pro re-homing him just because of that. Ironically, the 3 who experience mild allergies (myself, my brother, and my dad, not my mom) are the ones who want to keep him.
I talked to a trainer and dog adoption center owner today and they were against leaving him out overnight because, like you all said, separation from the pack will have long term effects on behavior. I relayed this to my family and they agreed in keeping a dog indoors overnight (as is Logan's case right now). So the shed (which BTW is a ridiculous $2000 from Home Depot and looks like a guest suite) is a no-go.
But about Logan being happy as "their" dog… after laying out my demands on ownership, my dad and brother didn't seem enthused. They're not excited about attending the next level of puppy/owner training, taking him to dog parks, nor exercising him morning and night. Though they enjoy having him around and are willing to help out sometimes, I don't think they'll follow through with the same full care I give Logan. So looks like it will be a re-home. To those who messaged me with interest, I will respond.
On a positive note, today Logan eased into meeting some friendly boxers/boxer mixes, wrestled with a husky, and ran with a German shepherd and a Doberman after over a month of not going to the dog park since his neutering. Guess the fixing helped... this was the first time he wagged his tail at bigger male dog strangers!