How to bevel a vertex corner that is between a circle and straight? Only way I can imagine doing it is by manually adding a circle and editing it into the shape. Which is slow. Same with knife project and that also creates other issues and is also slow. Maybe there is some magical hotkey with the bevel that can do it or something?
This pretty much exactly what the Spin tool is for .. 



In an orthogonal view of the circular arc you want to create, put the 3D cursor at the center of the new part-circle, select the vertex you want to extrude around it, and invoke the tool.

In your case, you might ShiftS snap the cursor to the two selected endpoints of the bevel, so it's exactly between them.

You can adjust the center during the operation by using the arrowheads, and the number of vertices generated in the sweep using the tool's Adjust Last Operation panel