I have not tried Gore-tex socks, but they seem not to cheap. I suspect they don’t vent too well, especially inside a boot that is not wicking to the outside. If you want to try in the narrow temperature range, just use plastic bags as your vapor barrier. Thin socks > VB > cushion socks. If you were wearing any type of thicker sock before you can find a combo that probably won’t size you up. That’s pretty cheap to try.
I’m also inexperienced and suspect of neoprene for the purpose you suggest. I have used wetsuits made from Neoprene for many years. Wetsuits fill with water and hold it against your skin. Your body heats the water and you feel more comfortable versus just being in the cold water. Any breeze and the water starts to cool off from the outside. Inside a boot, you would presumably be dry, until you sweat or have water seep in from the outside. I think the Neoprene will attract and trap the water in your boot and against your skin. It sounds clammy at best. Like wetsuits, it relies on your body heat to warm up the water. I just don’t see these drying out – ever, until it’s dry and sunny and you don’t need them anymore.
I converted to the non WP shoes years ago, but I do wish I had a waterproof/breathable boot for the times when it’s continuously wet and cold. You said your backpacking boots are still waterproof, but heavy. There are lighter boots that can be waxed or have a liner, and I’m sure they will work for awhile. Maybe the lighter they are the quicker you will need to replace or reseal them. I just started looking at Lems Boulder boot for this purpose. They are light and have a wide toe box, but probably don’t have the cushioning I need. I may also try some WP/breathable Keens – which fit me well or maybe Solomon, which I saw some good reviews for. They are not light by trail-runner standards, but for the times you know it will be wet and cold and unlikely to become dry and warm they may work.
If you are talking really wet and cold, check out a recent video by Shug on having warm feet when camping. https://www.youtube.com/watch?v=seqBY9VM1bs. It’s about sleeping warm and standing warm, more than walking warm. Also it’s not about lightweight camping for him in Winter, but he is a backpacker, was an UL backpacker, still relatively lightweight, and you might find it useful (and funny). Shug has many good videos – and he camps in the Minnesota Winter so I think he’s got the cred.