I am building a temperature and humidity controlled chamber
Since you are into building stuff, you might try "reverse engineering" that undocumented Chinese POS to make your own driver for the ultrasonic "mist transducer." This is a virtually impossible task for the neophyte builder because there is likely no information about the transducer or the microprocessor controlling it, which is likely labeled U1 on your
image2. It is usual practice for the manufacturer to file off any identifying marks, plus you have no idea how it is programmed, even if you find out what it is. Then there is the multi-layer PCB, in the way of identifying how everything is connected. Good luck figuring that out.
You gotta make all sorts of other stuff to control the temperature and humidity of your chamber, so why not make your own humidifier that works to your specification? All you need is that transducer from your Chinese humidifier and a driver, plus some "glue" circuitry to control everything. You were gonna ask us about the glue circuitry in another thread, right?
Or follow
@Harald Kapp's advice in his post #3, and simply parallel a set of relay contacts with the push-button switch on the Snow Mountain Humidifier. Problem with that is keeping track of the current state of the humidifier since that push-button has three "misting" modes: (1) first press turns on continuous mist, (2) second press turns on intermittent mist, (3) third press turns mist off. Sounds like, for your temperature and humidity controlled chamber, you just need on/off control.
One possible solution is to use the external humidity sensor, that provides power to the Chinese humidifier, as a "trigger" for a pair of 555 timers. The first timer is a delay timer to allow power to the humidifier to stabilize. The second timer is the "one shot" pulse, triggered when the first timer delay "times out," that you need to simulate pushing the button (once only!) when the Chinese humidifier is ready to accept it.
Then, if you do want to extend the button press functionality, it also has three LED illumination modes controlled by long presses of the push-button: (1) first long press turns on the LEDs, (2) second long press makes the LEDs "breathe" by presumably ramping their brightness up and down, (3) third long press turns off the LEDs. That all seems non-essential to your purpose and so can probably be ignored, as you mentioned in your first post.
We are curious as to what transducers you are using to sense temperature and humidity, and how these transducers are connected to control a heater and a humidifier. Will your chamber also include a fan to circulate the air inside of it? Will it have stainless steel walls to allow easy wipe-down and cleaning? Are you aware you can purchase such chambers in various sizes off the shelf at truly exorbitant cost?
I don't blame you for wanting to construct your own! Just be aware it can be a non-trivial task, depending on what specifications are attached to the temperature and humidity controls. Most common humidity sensors are slow to respond to changes in humidity, and may themselves be sensitive to temperature as well as humidity. The most accurate humidty sensor that I know of is a "chilled mirror" sensor. This consists of a thin first-surface mirror whose back side is cemented to a Peltier solid-state cooler, along with a temperature sensor that measures the front side (first surface) temperature. An LED or a laser diode shines on the mirror at an angle which reflects the light into a photoelectric cell. A control circuit lowers the temperature of the mirror until a film of dew sublimates onto the front-surface mirror. The dew disrupts and scatters the light reflecting off the mirror, causing the control circuit to allow the mirror temperature to increase and evaporate the dew. This is a negative feedback loop that oscillates closely around the dew-point temperature. It is a simple matter to convert the dew-point temperature into relative humidity given the temperature inside the chamber where the humidity sensor is located. Well, simple if everything is controlled by a microprocessor.