After experimenting with your CSS, I managed to achieve the desired result.
You could potentially try implementing something like this:
.DayPicker-Day--start {
color: #fff !important;
border-top-left-radius: 50% !important;
border-bottom-left-radius: 50% !important;
}
.DayPicker-Day--end {
color: #fff !important;
border-top-right-radius: 50% !important;
border-bottom-right-radius: 50% !important;
}
.DayPicker-Day--selected {
background-color: #C4C4C4 !important;
color: #1C1C1C;
z-index: 1;
}
.DayPicker-Day--selected::after {
content: '';
position: absolute;
background-color: #1C1C1C !important;
width: 100%;
height: 100%;
z-index: -1;
top: 0;
right: 0;
border-radius: 50%;
}
I suspect that the trouble may have been related to the z-index values.
If you're interested, here is a link to screenshots from my inspector tool which might offer some insight. I initially used #ccc as a placeholder and forgot to update it, so make sure to adjust that to the gray shade you prefer for your background.