I'm wondering if you're aware that WhatsApp's data store is on the phone, not on their servers. If so, how would you do away with the requirement that the phone stay online while using the web client?
Clean data and uninstall the app from your Android, uninstall the app and then install again. WhatsApp you ask if you want to recover some of your history.
"WhatsApp automatically makes backups every day at 4 AM and stores them in the WhatsApp folder of your Android phone."