The OP is a noble aspiration, but the real world will intervene at the most inconvenient times.
I will mention and then drop immediately that a BMX will become hugely frustrating on a 'family ride' both for the rider and for the others who have to keep slowing and stopping. Now to the meat of the post:
All my children (13, 17 and 19) are keen cyclists and have been since they were first able to turn a pedal. I ride with them all, but in all those years have ridden only twice with all of them (for pub lunches) and only four or five times with two of them together.
A 12yo boy may not want to ride with mum and a 9yo sister. I do not recall any of mine (who all get on well) ever feeling the need or desire to ride with a parent and a sibling. When they did it, it was for some other reason not connected with the ride.
It may not be helpful to say this, but mine seem to like riding with me because they've been doing it since they were tiny and it has become a 'normal' thing to do across years and years. But even the (just) 13yo does most of his riding alone. 9 may not be late to start with your daughter, but in terms of 'family rides', 12 might be a little late to start with your son. Apologies if this seems blunt or is wildly inaccurate.
My feeling is that if he likes the idea of a non-BMX bike, he'll find a way of letting you know. Is he so much smaller than you? Would he be able to ride your new bike?
I wish you well with this. As a family, we've had excellent times on bicycles.... It's just that it was almost always in groups no larger than two.