Viva, that's exactly my impression of your current coat wardrobe, and why I think you do need to invest in another coat (or two) for travel.
I suspect you and I may even have the very same Laundry coat, as your description sounds very similar to one I own. It's cute and stylish, good in the rain, but not all that warm.
Since I live in Boston there's always room in my wardrobe for lots of coats. The coats that I take most often on my travels include:
1. The AT red coat I purchased this past fall. It is not super warm but perfect for Spring and Fall, or very mild southern winter temperatures. It's a length that can work with dresses. It layers very well, and the color makes me happy.
2. A sleek lightweight packable puffer in 3/4 length. I just invested in a new one recently, found on sale, to replace my metallic blush puffer. I went with Bernardo brand, in a beautiful ink blue that has some sheen, with pink & orange zippers. In other words, a stylish puffer. It's so thin and lightweight, yet it is warm enough for all but the coldest temperatures. What's more it packs up small so I can tuck it into my carry on.
3. Burberry trenches, 3/4 length. I have a rain trench and also a black wool trench. The wool trench is very warm, and yes it is heavy. I have no problem folding my Burberry rain trench (size 6) and tucking it into my carry on bag.
I rarely travel with shorter jackets as I don't find them as practical to go over dresses, or to provide coverage if it should rain or be colder than anticipated. I find 3/4 length more versatile for travel.
I also nearly always wear one coat, and pack a second one. For this reason, I love packable puffers!
Finally I recommend investing in base layers made of silk or merino wool, or that heat tech stuff, so you can pop them under your clothes as needed for warmth. I always pack silk long johns -- they are my travel pj's. And yes a big pashmina scarf!