I looks like with firmware version 1.1.8.161917 does properly check for new firmware updates when the router is in Bridge Mode. I can't be sure but the below is what the messages log show (sysinfo.cgi) when a check for update occurs.
Aug 4 06:12:20 blk-mamba daemon.warn jnapcgi[12306]: Handled http://linksys.com/jnap/firmwareupdate/GetFirmwareUpdateStatus request in 0.013753 seconds...
Aug 4 06:12:20 blk-mamba local7.notice UTOPIA: config.olympus_api set_sysevent: update_firmware_now = 1 [ 44594 ]
Aug 4 06:12:20 blk-mamba daemon.warn jnapcgi[12307]: Handled http://linksys.com/jnap/firmwareupdate/UpdateFirmwareNow request in 0.015950 seconds...
Aug 4 06:12:20 blk-mamba local7.notice UTOPIA: autofwup.status update_firmware_now event : ,
Aug 4 06:12:20 blk-mamba local7.notice UTOPIA: fwupd.status Starting fwupd [ 44599 ]
Aug 4 06:12:20 blk-mamba local7.notice UTOPIA: autofwup.status update_firmware_now event : ,
Aug 4 06:12:22 blk-mamba local7.notice UTOPIA: fwupd.status firmware update done. StatusDetails:"" [ 44756 ]