I don’t think you will be able to accomplish this using the current firmware…though you could try using something attached to the rain sensor and some clever scheduling to make something work.
I would recommend checking greenhouse suppliers for a thermostatic solenoid, they are often used to cool areas in a greenhouse as well as help with humidity. They are usually not too pricey and would serve their singular purpose well when the health and happiness of your chickens is concerned.
Modifying the firmware could be accomplished without too much work but may be at the expense of other features if you would like to go the easiest/least complicated route. If you would like to add it as a feature perhaps using the current checking capabilities would be feasible. You could attach a thermister to one of the valve positions and run the mister zone based on certain parameters. Or use a digital thermometer of some sort with a pulsed output and read/interpret it through the sensor port. Personally I think a local sensor would be better than running based off of the weather, as the weather is only checked once per hour, though I suppose if you are modifying the firmware this could be changed as well. Just keep in mind the number of API calls you can make per hour and day is limited and could result in restrictions to the service if you exceed them too many times.
I do think it is a novel use of the system, I may try to add it to my forked version of the project. If I do make progress I can let you know, but unfortunately summer is a really busy work season for me.