As far as programming goes, fxload is included in most Linux distros and does not need any compilation. Although it is also weird and uses some vendor hex files for initial programming and i have not used it a lot. But I think it worked fine when I needed it.
There are also two of FX2LP devices and one EEPROM, so I guess the second one is loaded separately somehow.
These two chips may be connected to one EEPROM.
To program the EEPROM you always need suitable firmware running on the chip, and that's what vend_ax is. You also need fxload to upload vend_ax to FX2 RAM because the built-in ROM bootloader doesn't support EEPROM programming requests, but it supports downloading other FW to RAM. Once you have vend_ax uploaded fxload is able to overwrite EEPROM, but it cannot read it AFAIK, hence my recommendation to use fx2eeprom instead.
The python stuff above seems to achieve the same effect in one convenient package. Worth trying too.
This is the complete process for dumping 8KB EEPROM with my patched fx2eeprom:
# lsusb
...
Bus 001 Device 102: ID 0925:3881 Lakeview Research Saleae Logic
...
# fxload -I vend_ax.hex -t fx2 -D /dev/bus/usb/001/102
# ./fx2eeprom r 0x925 0x3881 4096 0 >part0
Readed 4096 bytes
# ./fx2eeprom r 0x925 0x3881 4096 4096 >part1
Readed 4096 bytes
001/102 are the bus/device numbers and 0x925 0x3881 are VID/PID.
edit
Your 24LC01B EEPROM is 1Kb only, i.e. 128 bytes. So you can read it in one go without patching fx2eeprom. It probably doesn't contain any firmware, only IDs (FW can always be uploaded by host drivers once the device is identified and that's the solution employed by most FX2 users), but if you are unlucky there could be some magic "password" which must be there for USBee's official software to talk to this board, or HW revision information, analog cal data or something like that.