Once you have opened the date picker, you should be able to send a click by using the value, as the value is the actual day value. So, if you modify the selector, it should work in theory.
The tricky thing is selecting a date in a different month. You'd need to build a functionality that extracts the current month, converts it to a date, and calculates how many times you need to move back to actually navigate to the month you want. Then click the back arrow that many times. The rest should be okay.
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