You do not need to seek a facility that is on the Bupa network. They publish the network because it's easier for you to go there, knowing the provider is already in the Bupa system.
If you get care in Belize you will pay out of pocket and then file for reimbursement from Bupa. Bupa will pay you efficiently and according to the policy. Be aware that the cost of care in Belize is so low that it's hard to exceed your deductible!