Have never heard of that (vinegar)…. have you been to the Vet to determine the cause of the skin infection?... Have you had a full thyroid panel done?..... before I would go to a home solution, I would make sure.... of what it could be or what is causing it... could be food... could be mites... and it is totally possible that the more bathing of him is making it worse... and this time of year Basenjis typically blow coat.. so it could just be that that is causing the shedding and dandruff... also, could be lacking stuff in his food... you should try adding Omega 3 to his food...
Also, your problem of picking him up should be addressed... there are lots of things that you can do to work with him to get him over this.... you should work on this.. as if there was a real emerency that you had to pick him up, or someone else... you need to prevent someone from getting bit by him...
You can make it a game... do things like first get him to accept you just picking up one foot at a time... and then rewarding him... then two feet... then just your arms around him like you were going to pick him up.... and so on.... While he might always "grump".. (and my male has done this for 15 1/2 years.. but we just call that "boy dog noises)... he has never offered to bite...