I gave up on the Enternet 300 software after creating 3 different profiles and reinstalling it once. The login's now saved into the router (it's a Netgear router/modem combination), which actually seemed to help a bit (it'll stay active for a tad longer).

The problem doesn't seem to be related to phone use.

I don't think the computer's trying to switch over to a dial-up connection; I have everything set up for a LAN connection and the dial-up modem's not even installed.