I hear you. I have a warmish winter coat that I only wear when I travel to Europe. If you're going to be outside a lot, think about thermals. You can wear them under your regular clothes (tops and bottoms) or instead of them. They are extremely light-weight, can be hand-washed and dry overnight, don't take up much luggage space either. If you aren't planning on keeping the coats after this trip, see if you can get anything second-hand, maybe even something like Craig's list. To keep my feet warm, I wear a very thick pair of wool socks over my regular ones. They are a nice comfy cushion too when there's a lot of walking. A light windbreaker under a coat is wonderful on windy days. A sturdy rain poncho is my choice for ease of packing, toting around and keeping me and my backpack dry.
You can get hats, gloves and scarves in Italy. I spent about €30 on a good pair of leather gloves at a market in Florence and I use them for driving at home in the winter, when the temps go down to the 50s. Other than that, I don't think prices for clothes in Italy, even on sale, will be cheaper than regular prices in the U.S. There should be winter close-out sales now in the U.S. If there aren't any suitable clothes in your area, do some web shopping. . . Macy's, Sears, Penney's, etc. Or be prepared to spend a lot of time and money shopping when you get to Italy. Generally the shops in the historic centers where you'll be touring will be the most expensive. The bargains are likely to be much farther out, where most Italians live.