The cpu is not the most power-consuming component of your device: the radio and display both consume more power than the processor. According to [1], the cpu typically accounts for only 10%-20% of power usage.
However, a secondary stream from the app to a different endpoint would be detectable, so there are other practical challenges to the GP's scenario.
Couldn't it be a secondary stream to the same endpoint but using a different keying (ie, encryption with server's pubkey) mechanism (the metadata then wouldn't be E2E encrypted, but that's not covered in WhatsApp's E2E security page)?
Is that feasible client-side without burning all your battery?