I have had stats that meet or exceed Star Seller status for a few months now, but I don’t seem to ever get awarded the badge. Do I need to formally apply for it?
No, you don't "apply".
You are missing 2 of the badges. Have you looked at the CSVs to see why not? You don't have the messaging or shipping on time ones.
No, you do not need to formally apply for it. Look into your CSV to determine why you are not getting it - you are missing info.
you are missing shipping and messages badges.
Did you have $300 in sales for the previous period?
We had almost no sales in December as we had to shut down the shop due to the Canada Post strike. But that was only for a few weeks. Otherwise yes we have very steady monthly sales well above $300.