Did you try to add an application setting WEBSITE_VNET_ROUTE_ALL set to a value of 1?
This setting forces outbound traffic through the virtual network and associated NAT gateway.
Without this setting, internet traffic isn't routed through the integrated virtual network, and you'll see the same outbound IPs.
https://learn.microsoft.com/en-us/azure/azure-functions/functions-how-to-use-nat-gateway