The SelectShippingMethod method accepts only the selected option formed according to a special template (which you specified correctly) and a dictionary of parameters for pickup points. Based on your code, it seems to me that you are passing the shippingOption parameter in the body, but it must be a parameter from the query string.
here is an example call on my local server:
curl -X 'POST' \
'http://192.168.1.172/api-frontend/Checkout/SelectShippingMethod?shippingOption=Ground___Shipping.FixedByWeightByTotal' \
-H 'accept: application/json' \
-H 'Content-Type: application/json-patch+json' \
-d '{
}'