I now see that, while my device (per 'dmesg | grep -B1 Broadcom') is a BCM20702A0, I was attempting to load firmware from a .hex file with "BCM20702A1" in its name. I had chosen BCM20702A1_001.002.014.0889.0897.hex because that was the only filename that turned up when I searched bcbtums-win8x64-brcm.inf for "21E6", which is my device ID (per 'lsusb | grep Broadcom'). 
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;;;;;;;;;;;;;RAMUSB21E6;;;;;;;;;;;;;;;;;

[RAMUSB21E6.CopyList]
bcbtums.sys
BCM20702A1_001.002.014.0889.0897.hex

[RAMUSB21E6.NTamd64]
Include=bth.inf
Needs=BthUsb.NT
FeatureScore=EF
CopyFiles=RAMUSB21E6.CopyList
CopyFiles=BcmBtRSupport.CopyList

[RAMUSB21E6.NTamd64.hw]
AddReg=RAMUSB21E6.NTamd64.hw.reg

[RAMUSB21E6.NTamd64.hw.reg]
HKR,,LowerFilters, 0x00010000, "bcbtums"
HKR,,%RAMPatchFileName%,0x00000, "BCM20702A1_001.002.014.0889.0897.hex"
HKR,,%RemoteWakeEnabled%,0x00010001,1
HKR,,%DeviceRemoteWakeSupported%,0x00010001,1

[RAMUSB21E6.NTamd64.Services]
needs=BthUsb.NT.Services
AddService=bcbtums,,BCBTUMS_Service_Inst, BTWSECFL_EventLog_Inst
DelService=BcmBtRSupport,0x00000200
AddService=BcmBtRSupport,0x00000800,BtwRSupport_Service_Inst ; SPSVCSINST_STARTSERVICE
___

So, I tried the wine trick using the Lenovo Thinkpad "Bluetooth with Enhanced Data Rate" software...

http://support.lenovo.com/en_US/downloads/detail.page?DocID=DS034630
http://download.lenovo.com/ibmdl/pub/pc/pccbbs/mobiles/gfwb06ww.exe

...and ended up in more or less the same situation. But, the Lenovo installer created an additional folder, though, called "LD" which contained a BCM20702A0_001.001.024.0156.0204.hex file that wasn't in the set from the Broadcom installer...so I figured I'd try that one out. But, alas, it broke things again. I was no longer able even to add the headset I've been using to test, and I ended up with the following, over and over again, in /var/log/syslog:
___
Sep 14 15:28:27 ... kernel: [  163.540986] usb 1-1.4: USB disconnect, device number 26
Sep 14 15:28:27 ... bluetoothd[1056]: Unregister path: /org/bluez/1056/hci0
Sep 14 15:28:27 ... bluetoothd[1056]: hci0: Set IO Capability (0x0018) failed: Invalid Index (0x11)
Sep 14 15:28:27 ... kernel: [  163.773364] usb 1-1.4: new full-speed USB device number 27 using ehci-pci
Sep 14 15:28:27 ... kernel: [  163.869583] usb 1-1.4: New USB device found, idVendor=0a5c, idProduct=21e6
Sep 14 15:28:27 ... kernel: [  163.869595] usb 1-1.4: New USB device strings: Mfr=1, Product=2, SerialNumber=3
Sep 14 15:28:27 ... kernel: [  163.869601] usb 1-1.4: Product: BCM20702A0
Sep 14 15:28:27 ... kernel: [  163.869605] usb 1-1.4: Manufacturer: Broadcom Corp
Sep 14 15:28:27 ... kernel: [  163.869610] usb 1-1.4: SerialNumber: <serial_num>
Sep 14 15:28:27 ... mtp-probe: checking bus 1, device 27: "/sys/devices/pci0000:00/0000:00:1a.0/usb1/1-1/1-1.4"
Sep 14 15:28:27 ... mtp-probe: bus: 1, device: 27 was not an MTP device
Sep 14 15:28:28 ... kernel: [  164.670213] Bluetooth: firmware loaded
Sep 14 15:28:28 ... bluetoothd[1056]: input-headset driver probe failed for device <mac_address>
Sep 14 15:28:28 ... bluetoothd[1056]: Adapter /org/bluez/1056/hci0 has been enabled
Sep 14 15:28:28 ... bluetoothd[1056]: hci0: Get Connections (0x0015) failed: Not Powered (0x0f)
___

All of which probably means that I'm off-topic, here, huh?. :) But still. Anybody got any suggestions for how I can do a better job of finding (or converting) the appropriate .hex file?