Simulating resistance for buttons

It looks to me like you did not completely understood the schematics.
You connected pin 9 to what looks to be a pull up resistor.
That is actually what is does, but only until a key is pressed.
In case a key is pressed, it is part of a divider.
The keys have an array of resistors, and that key that is pressed (in case multiple keys would be pressed, the one most to the right will set the value) connects to 0 volt (GND).
So in case the 5th key is pressed, you can add 1K5 + 1K + 1K + 680 + 470 (i'm guessing because i can't read all values) so that would become 4650 or 4K65.
You will have an array of resistors forming 4K65 and an other resistor of 4K7.
These two values have 5 volts across, and at their joining point you will have about 2,486 volts.
This is quite close to a value actually measured by yourself.
You cannot just make this measured value and insert it instead of the resistor/key.

I would just connect pin 9 to the input of the unit, without this 4K7 resistor to 5 volt.
Thus, put your analog output directly to the input (that arrow pointig out) but not to the existing resistor.

You still need to make the PWM output an actual analog output, because else it's still unlikely to work.
And of course you always need to connect the GND of your Arduino, to that of the unit you want to control.