I think the reason many Basenjis hate these Varikennels is that they restrict vision - the puppy can't see what is going on and feels vulnerable. I only ever used one when shipping a puppy (accompanied by its new owner) by air.
Open wire crates allow the pup (or adult dog) to see around him/her.
If only breeders would help in this respect. I took a wire cage to the breeder of my new puppy with the request he become accustomed to it. The crate was left, open, with the puppies from the age of about 6 weeks. The whole litter played in it, slept in it from time to time, were individually fed in it on occasions and I have a lovely photo, sent by the breeder, of my boy curled up on his own, fast asleep in the crate. Now all six puppies have benefitted from this early crate training, at least one breeder will repeat the exercise for all future litters !
It doesn't involve the breeder in extra time, just a bit of extra space in the general play area. I always did it - so even if the puppy was taken away by car, a crate held no fears.
In the present case, I suggest binning the Varikennel and getting a Basenji-sized wire folding crate that you can put away and get out at need. Put it at first where the puppy plays, leave it open, put toys in it and encourage him to come and go as he wants. You could even put all his toys in it so he has to go and collect them when he wants to play. Use it like a toy-box. Be patient, make going into the crate a game, get down on the floor with him and play, throwing his toys into it to get him to follow them. Don't shut it at first but then when he is used to going in and out you can shut it for short periods and eventually for longer if / when he actually goes to sleep in it. (make sure it has a lovely comfy blanket in it !).
Also, PLEASE tell us his name and the names of his parents and date of birth so I can add him to the database of Basenji pedigrees on line - the address is in my signature block - so you can have fun seeing who his great great grandparents are