I am so sorry your baby is not feeling well. He is a baby at 5 months.
Pale gums can be indicative of anemia.
The thing that concerns me is if he is not having bowel movements he may have swallowed a non food item
I am really not happy with a vet that says a tiny dog that is vomiting is not sick,That is ridiculous!
He may have a bad tummy ache. I would pick up any dog food.Try a small amount of boiled chicken with all fat drained off, chop meat , or baby food with no onion, beef, turkey, etc
Go very slow so he does not throw it up.
Small dogs get dehydrated and hypoglycemic very easily from diarrhea.
Add unflavored pedialyte to his water 50/50 to hydrate him and balance his electrolytes.
Put a drop of pancake syrup on his tongue to raise his blood sugar in case he is a bit hypoglycemic. That should perk him up.
Do not allow him to drink a lot at a time If he throws up water, he can lick ice
Keep him on that very light diet of boiled chicken or chop meat for 48 hours after the vomiting stops.
Also give him one teaspoon a day of plain yogurt to keep "good" bacteria in his system.
If he develops any abdominal pain, bloating, constipation or cannot keep anything down he needs to be seen by a DIFFERENT vet.
Please let me know how he is doing. I will always be here for you
I hope your baby feels better very soon..