How to keep the internal URL in browser address bar and point to external website

That's a fairly basic problem and I recommend you look into LTM Local Traffic policy feature for the most performance-efficient solution. However, you can also use an iRule provided below. Of course, you will also need two DNS entries - one of which is used by the general public, the other for your private LAN users/employees.

when HTTP_REQUEST {
  if { ([HTTP::host] eq "abc123pqr456xyz789.azurewebsites.net") && (not ([class match [IP::client_addr] eq "private_net" ])) }{
    HTTP::header replace Host "honest2.company.com"
  }
}