Line loses power when light bulb is put in
To me this sounds like a bad connection. My first look is for backstabs prior to the switch or at the switch, it can also be a broken wire sometimes a Knick on the wire when striped the wire can break over time, and last a wire not fully in a wire nut. 
All of these can show full voltage without a load but once a load (the light) is added it opens. I see this most often with backstabs the push in wire connections , I have found a few broken wires and a few more loose wire nuts over the years but most often a bad backstab. 

Although I have found this problem hundreds of times on outlets I have found it a few dozen times fed from an outlet on a lighting circuit, 
Usually lighting circuits are not as heavily loaded but in a daisy chain feed the bad connection is usually close to the switch a outlet that feeds the switch within a few feet. Or the switch itself. 

You mentioned a double pole breaker so it might be a multi wire branch circuit, but since both hot to ground and hot to neutral voltage are good I would be looking for a bad hot , black or red wire.