John,
What are you attempting to do, to issue a GET or a POST to push data to a server?
From what you say, it seems you are using DHCP, and that the Arduino seems to be getting an IP address and then falls over, but have you tried not using DHCP and instead, specifying the IP parameters?
Could putting debug points (serial prints) in at strategic places help you, even in the library?
Could it be something in your router's configuration?
Are you using the latest version of the IDE together with the supplied libraries, not mixing compiler version with library versions?
It appears you have delved deeply into the problem and have an understanding of the EthernetClient library.
You mention;
Accepting incoming connections from my browser
Can you explain this more clearly, as I don't really understand what you mean here?
I am not sure what else to suggest at this time.
Paul