Our Kaiser is feed a raw diet and only natural treats and had started to chew his feet and scratch his ears and lick lick his hernia(seemed to affect his skin where he is white and where he has no hair and it is pink skin which also came up with a few little lumps here and there), this started after we went to the dog park twice in one day, which has long grass which he just loves to bounce through, hadn't noticed it so bad until that day, other times he came back a little bright pink but it would go and there was no chewing and licking, I thought this was due to the heat and running around. Found we also had some wandering dew coming through from next door but that I had cleared through betadine soaks and this was cleared previously to the park incident and mainly affected the two front paws not all four like after the park and he didn't chew as much with the dew as well. To get him over the eating himself alive episode I gave him half a tablet twice a day of Histaclear which is the same as Razene and Zyrtec which are all allergy meds. and kept up the betadine soaks and he has cleared up after 2.5 days, now I will test him out at the park in the long grass will also have a good look to see if there is any wandering dew about and see if he reacts and then I will speak with our holistic vet as he is due for a titre and see what she thinks. So pretty much thinking he has a contact allergy any chance your basenji has come in contact with something similar.
Sorry wanderind dew also called wandering jew, wandering willie it has a few names, just search one or the other and you should get offending plants.