Hi there. I had a similar problem at my old job as I did field support for customers via dial-up and also had to access the trouble ticket system on the LAN.
What is happening is once you dial-up to an ISP, Windows tries to route any and all traffic through that dial-up connection and it kind of "forgets" about the NIC in the PC and any LAN information that it has stored.
To get around this "glitch" in Windows 9x you can do the following on the PC that needs to dial-up:
1. Create a batch file
2. In the batch file put the following:
route add 192.168.1.2 MASK 255.255.255.0 192.168.1.1 METRIC 1
3. Save the batch file and have it launched automatically through your startup group or create a shortcut on your desktop that you can launch manually.
What that route add is doing is adding the 192.168.1.2 (which is the other computer on your network) to its routing table, and you are telling it to use your subnet (assuming its 255.255.255.0) and telling it to route any requests for the 192.168.1.2 computer through your 192.168.1.1 NIC on metric 1. Metric 1 will be your initial network configuration when you boot up. Once you dial-up to an ISP, the dial-up connection becomes metric 2, etc.
Understand what I am saying?
This will take care of your problem once you get the batch file setup correctly as it solved my problems in the past.
I know this is a little vague and I did make assumptions on your IP and SUBNET masks, but if you plug in your correct information you should be all set.
Let me know if you have any trouble.
Good luck!