For conversion of an image to DXF, I would use Adobe Illustrator. I am sure there are other programs that can do the same. When faced with a similar problem of converting a scanned image to vector, I used the AI auto trace feature, exported to my CAD in DXF, then retraced manually and fixed any dimensions that were a little off. Relative dimensions were generally fine. I had to shrink or stretch the entire drawing/component to get it just right . That seems like a lot of work, if your PCB program will simply accept a bmp that you can trace.
It seems that three common PCB layout programs accept a BMP for background tracing. Are you sure your program doesn't have that capability too?
John