You may be in a situation similar to mine. We are testing against a custom date picker written in HTML/CSS/JS… We cant use send keys either since the input is disabled. Instead, we have to simulate the user clicking/tapping through the date picker so our Selenium goes:
find xpath to open date picker -> click it -> find xpath for button to move backwards one month in the open date picker -> click it -> find xpath for the date we want -> click it
Its a pain but we were able to wrap that in a reusable method that accepts a date to find as a parameter.