Hi Pawla,
While there's no harm in changing his food, I've found Earthborn in general to be a great product. Since Meadow Feast is grain free, maybe try him on a an Earthborn product that isn't grain free:--the pea/bean/lentil flour often used in grain-free dog food can be problematic for immune issues. If your budget allows, why not try him on a commercial raw diet? Turkey is an anti-inflammatory meat, and usually a more reasonably-priced choice. Try to at least stop him from licking his foot after meals during this experiment--easier said than done, I know!--since the bacterial load in his mouth might ramp up slightly after eating. In general, licking will cause infection--it may be that the rash started as an allergy then became an infection from licking, so maybe try a soft collar. I taught my BRaT boy "Leave it" because he was a nervous licker, and it actually worked, but you might need a soft collar. I've jury-rigged them from a 3-4" wide piece of stiff-ish upholstery foam held in place by the dog's collar--it stops them from bending their neck enough to lick, but is still comfortable.
That said, basenji skin rashes usually start from environmental issues. Dust/mold/mildew from central heat or air; Cleaning products used on bedding & rugs or heavily scented products spray products, off-gassing from new rugs or carpets; Even residue from baking soda or other carpet cleaners can cause a rash if not sufficiently vacuumed. Toilet bowl or shower cleaners may not be overtly toxic, but the VOC's and chemical compounds definitely weaken the immune system over time if the dog drinks from the toilet or shower. Outdoors, look at what surfaces he's walking on/through--a dog I walked at the SPCA got a rash between his toes from tough brush; Herbicides or pollen in grass could be an issue. Do you have Foxtails in your area, or did they exist near his former home? Check with your BRaT coordinator, since that could be causing his foot irritation. Maybe walk him in a set of rubber sole dog socks like Sport PAWks or Grippers to see if that helps--my b's prefer them to boots.
Finally, it may also be that stress from his previous situation eroded his immune system, which may need help getting rebooted. Raw diet will definitely help with that, and probiotics are vital: I give stressed fosters organic goat Kefir to increase their gut flora, it's like yogurt on steroids.
Organic cow milk kefir is often sold in grocery stores if you can't find a health food store that sells goat kefir. Probiotics from the vet (made by Purina) made one foster sicker, which is why I now use Kefir. It--and human grade Omega 3 supplements--are my go-to "meds" for inflammation and allergies.
Reducing cleaners etc mentioned above will also help, if you use a lot of them--I switched to homemade vinegar or lemon based spray cleaners and baking soda scrubs years ago (foster dogs teach me so much!) and haven't gone back. If you have grandkids, it's healthier for them, too.
Sorry for the epic post--I've been in your shoes and know how upsetting it is, so I wanted to share my experience. Feel free to contact me if you have questions, and thank you so much for adopting a BRaT 
