Can I install a brick with a corner missing by filling the gap with mortar?
Yes, you can backfill the broken brick with mortar and use it. Masons have done just that since they were using natural stone. 

An occasional mortar joint of an inch and half isn't the end of the world. Foundations that step vertically over rough footings often have large gap fills. The key is good mortar quality, and to understand that it may look odd. Avoid non-standard layout if you can.