if you can actually get on the internet with your router i would say its not an issue with mac reservation.
as to why your router keeps droping the connection i can't tell you, you can try updating the firmware to the most recent version and reseting your settings to defaults to see if its works any better.
if you wanted to test out his mac theroy it is easy to fix go to your router page from your main computer (the one you have hooked up directlly to your modem and it works fine) click setup, manual internet connection setup, clone your pc's mac address, save settings and reboot both your modem and your router.
no need to call your isp