I thought someone with direct experience might reply but as they haven't, I'll contribute my thoughts - I've followed some pilgrims who walked the route in January. It can be done, but you need to be ready for solitude and cold.
With three weeks, you might have enough time to reach Merida - If you enjoy exploring places on the way, you could consider doing some sightseeing in Malaga, Cordoba, and Merida. (Baena, Medellin, and Don Benito are also nice towns to look around as you pass through). Merida would be a good place to finish. I've heard that the Basilica of Santa Eulalia in Merida is one of the earliest recorded pilgrimage sites in Spain.
You can expect to have a solitary Camino. The Malaga route doesn't go as high in altitude as the route from Almeria but temperatures in Baena, Cordoba, and Merida may hover from around freezing point up to low double figures in December. (Check climate graphs online). Some albergues may lack heating, so you'll need a winter sleeping bag. I believe that they should all be open, though. You could contact the Malaga and Badajoz associations via their facebook / web pages to confirm.