Well, it could work… but I’m not sure why you would want one. OK let me back up.
One of our customers asked if there were a cellular booster that boosted Bluetooth signals instead of cell signals. That way, you could leave your phone on the charger, or leave it where the cell signal is good, and walk around with a Bluetooth headset. Sadly, I do not know of such a device.
The whole point of Bluetooth is actually that it is a short-range protocol. Part of the reason Bluetooth is hard to hack is that you really need to be pretty close to whatever it is that you’re trying to intercept. This isn’t like someone from three buildings away can pull your Bluetooth signals. Realistically Bluetooth will work pretty well within about 10 feet and while theoretically that does mean someone on the other side of the wall could hack you, that’s not as bad as Wi-Fi, which covers quite a distance and is easy to hack if you have a password that’s easily guessed.
If Bluetooth signals did carry for a long distance, there would be an increased risk that your keyboard, headset, mouse, or any other device could be easily intercepted. I know, no one thinks it’s going to happen to them, but this sort of thing could be pretty scary if you think about it. Someone who could hijack your keyboard could take control of your computer quite easily and someone who could hear all your calls could literally ruin your life if they wanted.
So no, there’s not a Bluetooth booster that I know of, and that’s probably a good thing.