Hello.
I made some testing and it seems that you were close on the CSS that you tried.
I sugest you to try the same thing with the following CSS selector:
.animated-label-input .form-control[data-input]::-webkit-calendar-picker-indicator
By using this and "display: block;" I managed to make it always visible. I did not went much deeper than this but feel free to ask if you need anything else.
Best regards.
Edit:
I tried to make it appear only when active and it works on my tests with this CSS. Try it and tell me if it works out for you:
.animated-label.active .animated-label-input .form-control[data-input]::-webkit-calendar-picker-indicator {
display: block;
}