When you applied you were supposed to tick which address your bill would be delivered to. if ticked correctly then insist on the manual change as it was a mistake of Globe. If by any chance you did not tick it or it was ticked wrongly then you would have to inform Globe in writing about the change of billing address
Quoted from Terms and Conditions:
The Subscriber must inform Globe Telecom in advance and in writing of any change in the Subscriber's billing address. Globe Telecom shall not be responsible for the consequences of the Subscriber's inability to receive and pay his bill which results because of the Subscriber's failure to timely notify Globe of any change in address.
I had my billing address change through online portal..you can try also calling the hotline or their twitter. The request was smooth as long as you can provide the personal information they needed.