Use AirPort as Time Machine whilst using internet
You could connect your AirPort Time Capsule via Ethernet and turn off the box’s Wifi altogether. This way, it will become part of your existing (wireless?) network and you can back up and access the Internet at the same time.

Any Airport device can be set up in “bridge mode” and they will join a network like any other computer as opposed to making their own network and requiring correct routing setup.