Before going any further, I'd clarify in my mind the intention of buying this bike. Is it for you, or your son?

Buying a bike to help him get into cycling is an admirable goal, but as you've already found, he (rightfully) doesn't care for something that doesn't fit.

A shared bike between people of two different statures is asking for trouble. It will be too small for him and too large for you. Beyond comfort issues, there are repetitive-motion injuries to consider. I have a feeling a shared bike will turn into an albatross.

Like Stapfam, I've found the top tube is by far the more important number than "frame size", which usually is determined by seat tube. Two of my bikes are 56s, one is a 58. All three have the same top tube length, within a quarter-inch, (5mm or ½cm) and all three fit well.

You can't go by overall height because body proportions may be different. My father is 6'-3" I'm 5'-10" He has a 32" inseam, mine is 34" Despite my being shorter by five inches, I have the longer legs.

Does that mean I'll ride the "bigger" bike? Not necessarily. He will ride a longer top tube than me, and I'll ride a longer seat tube, or more likely, I'll have more seatpost showing on a "smaller" bike.

As for "frame size"? Depends on the framebuilder. LeMond, for example, was known for longer than usual top tubes for any given "frame size".

If you're buying for yourself, measure the top tubes of your existing bikes. The measurement is always made center-to-center (center of the head tube to center of the seat tube) and parallel to the ground, starting from the top tube's level at the head tube if the top tube slopes. Determine which you prefer and buy the closest top tube size to what you like. You can fine-tune with the stem.

If you're buying for your son, get him to a fitter first.

As for the BD bike, not anything wrong with it, except for the lack of rack and fender mounts. They're both a matter of personal need. If you don't need them, it's not an issue.