We bought a Yorkie in May for my daughter's birthday. She too objected to the rain, and it was potty trouble all over. I spoke to my vet for suggestions when we were traveling to my sister-in-law's for the weekend - she has nice new carpet. My vet said that the dog was cold and to put on a sweater and booties. Apparently little dogs chill easier and are more persnickity than bigger ones. I drew the line on booties for bathroom trips.
We tried the sweater and it worked. Now sometimes she will still hesitate when it is cold or raining out but she will still go and get her business done when she really has to.
Maybe the sweater will be warmer than the rain coat. Also if the raincoat is bulky or obtrusive it could be scaring the puppy.
C.