I have a chi-weenie also and he get a bath once a week, maybe it's the shampoo. I went to my local pet shop and got something for sensitive skin they hair tend do be a little dry at time they will advise you what type of shampoo you should buy for your dog. Good luck they are great dogs. Hope this advice help you.
Check with your vet about Thyroid hormone test. it is a simple and quick blood test. Our Dachshund suffered from low thyroid hormones all his life, but it was a simple matter of giving him his pills... easy since it was since puppyhood. Hypothyroidism causes hair loss and is fairly common in dachshunds, so I would imagine common in chiweenies as well.
YES! Do not bathe a dog more than once a month in cold weather climates. In Summer, I bathe him every other week.