I don't have a recommendation for a board but if the fan is going to be powered from the battery then its power consumption will swap the consumption of any reasonable board to control it, and will be the main limiting factor for battery life.
Among you listed, ProMini is the smallest.
But ProMini don't have USB (for programming this), so you need USB serial adapter for program.
Yeah, that's why it's small.
If you don't like that, I think you should choose Nano is better.
Nano can be developed by connecting it to a PC via onboard USB by only itself.
Sorry to say but a pro-mini is not the smallest.
Check our "arduino beetle". I have used a number of these and they are great. There is a version with USB tabs on the PCB so you can plus it into the PC USB port to program.