Change session shipping country on product page/collection page

With geolocation the user sees the prices based on his location, as the session is pre set and the country in checkout is already preselected with what the geoip service is giving us back.

Outside EU = no tax
Inside EU = apply tax rates
Currency based on the visitors country

Unfortunately we had bad experience with GEOIP in the last half year and received bad feedback, so we want to give the customer the possibility to change the shipping country at any time via a sticky dropdown. On collection pages, on the home page and on the product page.

We have found the hooks:
woocommerce_customer_default_location
default_checkout_billing_country

But neither achieves the desired effect of changing the shipping country completely in the session, exactly like when I go into checkout and change the country.

Any tips from your side on how to implement this?