The ease of doing this will depend on your level of comfort with flashing firmware upgrades.
What you are trying to do is create a wireless bridge. The easiest way to do this is to purchase a device made for this purpose, like the Linksys WET54G. Such a device will connect one or more wired devices to an existing wireless network.
Alternatively, your WRT54G router has this capability, but not out of the box. You would have to update the firmware to an open source firmware called DD-WRT. (There are other firmwares that will do this, but I have found DD-WRT to be an excellent, easy to use solution.) The process is very easy, although as with any firmware flash, there is a minuscule risk your device could be damaged. I believe it would also void the device's warranty (if any).
You would also have to switch the routers in your diagram, hooking the BEFW11S4 into the primary position at the switch, and using the WRT54G router to extend the network.
Once the firmware has been flashed, it is a simple matter of going into the wireless settings tab and changing the router from access point mode to "repeater bridge" mode. This has the added advantage of still allowing the WRT54G to extend your wireless network and accept client connections. Also, DD-WRT provides a myriad of options like boosting your wireless signal that are not present in the default linksys firmware.
The exact version of the DD-WRT firmware you would need depends on your hardware revision number, which should be located on the tag on the router that has your serial number and other info. Any hardware revision BESIDES 7.0 should be supported. Use that number to look up the proper firmware version that you will need. Be sure to update the firmware from a computer that is wired physically to the router.
Let me emphasize again that you should ONLY attempt this if you are comfortable with the instructions found below. DD-WRT is a very user-friendly firmware install, but using the wrong version of the firmware will most likely render your router useless for anything but a shiny blue doorstop.
You can find instructions for the DD-WRT install process at
http://www.dd-wrt.com/wiki/index.php/Installation#Identifying_Your_DD-WRT_Firmware.
You can find the table of supported devices and firmware versions at
http://www.dd-wrt.com/wiki/index.php/Supported_Devices#Cisco_Linksys_.28Wireless_a.2Fb.2Fg.29Finally, you can find (very) detailed instructions for bridging at
http://www.dd-wrt.com/wiki/index.php/Repeater_Bridge. But in short, all you need to do is set the WRT54G to 'repeater bridge' mode under the wireless>basic settings tab, wireless mode dropdown menu.
