In the past I have sold a puppy (three in fact, over a period of at least 20 years) to someone with an allergy. It worked fine and she bred some very successful litters herself over the years. But I think it depends on the allergy, the severity and it would be wise to visit breeders and mix with their dogs.
See how you are affected before embarking on a very long experiment. Basenjis live a long time.
Read all you can too - there are loads of articles on my website, including some devised to deter people from getting the wrong dog if a Basenji is not for them.
You don't say what kind of special needs your children have. I am loath to sell a puppy into a home with very young children or unless I can be sure that the parents have done lots of homework and realise a Basenji is a very patient dog, up to a point.
But when playtime is over, and the dog wants its own space for a while, it is no use whatsoever allowing the children to persist. There will be a warning growl and the kids MUST know that is a 'Stop Sign'. Parents must take the dog's side, if necessary against the children, and never assume the dog is in the wrong. One must be aware that the children may ignore warning signs.
Email me privately (there is an address on my website) and I will send you a list of Breed Club Secretaries in UK. They should know who is breeding in your area.