Sadly, we do not have a DNS server in our network...
I don't have any idea what to say.
I've never even heard of such a thing. An IP network includes a DNS server. It's just a given. Without one, you can't do name resolution. So nothing will work except by IP address.
I mean, are you still using HOSTS files or something? Yikes. I'd probably quit and look for a new job. :-P
You're not going to make it work with NetBios resolution, of all god-forsaken things. Not while making it still accessible to the outside world. You could add the entry to the HOSTS file on every computer in the network. I've done that for short term hacks, but not on any kind of large scale. I shudder at the very thought of it.
In theory, you could write a filter that would recognize the IP address of the requesting system and return a different value for those two settings depending on whether the IP was local or not. You'd want to add your filters to 'option_siteurl' and 'option_home' using the add_filter() function. Then you would look at the incoming IP, figure out value what you wanted to use, and return it. That would probably work.