So, it looks like this may have been a custom device made for someone at some point, which is why the model number is a little off. That's also why 123Scan wouldn't allow me to update the firmware.
I did some spelunking and found the NBRACAAM-NBCACAAR.DAT firmware file at C:\ProgramData\123Scan2\Plug-ins\LS4278\LS4278-COMMON MODELS-S\019\LS4278-COMMON MODELS-S-019 on my machine. Then I used the sample app from the Windows Scanner SDK at C:\Program Files\Zebra Technologies\Barcode Scanners\Scanner SDK\Scanner SDK\Sample Applications\bin\ScannerSDK_SampleAPP_CSharp.exe and selected the listing for the cradle under Connected Scanners.
I switched to the Advanced tab, clicked the Browse button to point to the NBRACAAM-NBCACAAR.DAT firmware file, and clicked Update. Figured if I bricked it, an old used scanner isn't a huge loss.
Took a while to update, then I clicked Launch and it rebooted with scary red lights, but eventually came back.
Once it came back with the new firmware, beeping now works correctly (when sent to the scanner, not the cradle).
Long story short, find a way to update that firmware!