I have a similar configuration working flawlessly.
A Teensy++ with a WIZ812 Ethernet module connected to a TP-Link Tn-W703n pocket WiFi router
The Pocket router comes configured as an AP and connecting to it works out of the box.
Configured in client mode (I guess bridge mode is a better description) sending and reiving OSC commands to and from the Teensy++ through my AirPort Extreme Network Router (AP) worked the first time.
For programming the Arduino IDE is used in conjunction with the Teensyduino plugin. And I am using the Arduino DHCP/Bonjour libraries and ArdOSC in conjunction with TouchOSC.
I would believe if your TP-Link is correctly configured for bridge/ client mode the DHCP client on the Arduino should receive it's configuration from the DHCP server on your Linksys AP and not from the TP-Link. Otherwise the TP-Link is not really working in bridge/client mode but as a router. You could get that to work too by adding an entry for it to the routing table of the TP-Link router.
If you got it to work with a Notebook instead of an Arduino, however, There could be some compatibility issue with the software on the Ardoino and the TP-Link. Perhaps you could try your luck with the mentioned DHCP/Bonjour library http://gkaindl.com/software/arduino-ethernet
IIRC you may have to chage ao few little things to get it to work with the Arduino IDE 1 but that's easy enough to do. One of the sample scripts is a little Web server.