Does Contwoyto Lake empty into both Bathurst Inlet and Chantrey Inlet?
According to "Evaporation from a Small Lake in the
Continental Arctic using Multiple Methods" Nordic Hydrology, 27, 1996, 1-24:


  Drainage from Contwoyto Lake occurs through the Burnside River and Back River


So, yes, to Bathurst Inlet via the Burnside River and to Chantrey Inlet via the Back River.

You can trace out the two paths on the following topo map: http://en-ca.topographic-map.com/places/Contwoyto-Lake-947046/

The Back River outlet is at the south end of the lake and flows through a series of smaller lakes.