"It will quickly be drained"? Have you actually used the app?
I do use it, and the effect on the battery is negligible. WhatsApp has to run a background service in order to alert you to new message, so it's in contact with WhatsApp's servers anyway. Using WhatsApp Web on a computer makes virtually no difference to the amount of communication that your phone does, and has no measurable impact on your phone's battery. And this is direct experience of using it for several years on more than one phone.