From one Amy to another, welcome to PT and congrtas on the new lab.

My Mom got a puppy from a rescue group last May and he came with Kennel Cough. My now 1 yr old lab mix caught it from him, as did my mom's now 13 yr old beagle/poodle mix. All three had a bad cough, but not the nasal discharge. If the meds don't seem to be helping, maybe you should consider a second opinion or I assume there is a way to test for distemper. Force your vet to do it. If you are paying, they shouldn't deny it. Sometimes the extra cost is worth the peace of mind.

Hope your pup feels better soon!