Yes, basically Bluetooth is already almost too complicated to do well, and you would rather it were more complicated still? And like you say applying a convolution is computationally trivial especially compared to decoding a lossy audio codec and running a Bluetooth stack and antenna, which the headphones are already doing. Offloading this to the device is going to make absolutely no difference to battery life while increasing complexity and unreliability, while also restricting what processing can be done to static convolution with an impulse response. There is no reason to do this on the device.
It certainly would make a difference to battery life. Bluetooth connections as well as decoding are done in hardware using commodity chips that can't do much else. Adding additional audio processing hardware will increase the complexity of these chips which translates to higher prices and lower battery efficiency.
Remember, some of those devices have 20mA of battery. The codecs already have to be made easy to decode.
There is also absolutely no need to limiting processing to a static convolution with an impulse response. That's just the only device-specific processing you have to do. Headphones are minimum phase devices, so except for things like distortion, they can basically be described by their single impulse response.
For the rest, like spatial audio, or EQ, or anything of the sort, there is no need to do it per headphone, it's the same for all of them.
