I would have said that Bluetooth is completely legitimate and doesn't present a load to either the system or batteries. I then did a quick inventory of what I had.
Code:
Bluetooth "classic"
-------------------
5 Way Clicker - $20 AV style remote
Dual Media Player - $20 car stereo
Anker A7726 - $25 keyboard
Bluetooth BLE (commercial products)
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AB Shutter3 - $1 Chinese selfie clicker
iRig BlueBoard - $99 foot board
Bluetooth BLE (my stuff)
---------------------------
TestBed
UniPod
YellowBoard
KeyFob
I was surprised that a little handheld clicker used "classic" Bluetooth. I didn't think that the keyboard would be "classic" either.
Bluetooth "classic" tries to maintain a more constant connection and uses more power. Bluetooth "LE" (Low Energy) doesn't. It's not particularly easy to tell the difference between the two as a consumer. "3.0", "4.0", "5.0" does not tell you if it's LE or not.
If you're really concerned about power, try to find a Bluetooth keyboard that is LE. The power difference is not that big, but it might help. Check to make sure that there is no problem with latency (although the eink itself is probably the biggest factor).