Hmmm....tricky! And you have power-cycled your cable modem each time you reconnect the router? Of course you have but I had to check. Maybe someone else on the forum who actually has one of these might be more help at this point. Alternatively I could give you some pointers to setting up a Linux box as a router/firewall/wi-fi access point but I expect you don't really want to go down that route!
Actually...one thought - does the router need a crossover cable to connect to the modem? It's a long-shot as most kit these days is auto-sensing and will cope with either. Help!