I bought my 10 pound Cooper a jacket last year at Walmart and absolutely love it! I bought the wrong size and I never return items, so I gave it away. Here is a photo of it that I made to show people if they were interested. I then bought the correct size and am now looking for another one for Max.
I don't remember the brand, but I'm assuming Walmart might have it back on the shelves soon. It's very lightweight so that Cooper can wear it indoors if he wants. But it also has a super warm interior so he's nice and cozy outside. It's also perfect for the rain. There are 3 piece of velcro that allow quick on/off.
Here is a photo
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And attached is Cooper wearing it with his pajamas on
I really think Alex would be very comfortable in this. I'm going to really start searching for it and hopefully can direct you better.