We had somewhat similar issues with Jengo. Months after getting him we noticed red spots on his belly. thought maybe they were bug bites. Bathed him and inspected for fleas, etc. My wife read up on it and decided it might be a food allergy. She decided to eliminate grains altogether. That seemed to correct the issue. But, after some time he got picky, so we'd have to explore more grain free foods. It was a bit of a struggle. Still not certain if grain was the issue or not. Speaking to my vet, she mentioned that super high protein diets were all the rage for awhile, but that vets later started seeing more heart and artery problems. She doesn't like them anymore.
I'd never really known about Basenjis and veggies when we had Jengo. I wish that I had. Both new dogs get plenty of daily of unprocessed and processed protein. Kibble is mostly protein, dehydrated meat patties, raw meat, scrambled eggs But, they also get veggies too. Pumpkin, coconut oil, broccoli, brussels sprouts, cauliflower, etc. We change it up every now and again to keep them guessing. Veggies are also great snacks for starving dogs like Logan.