Main Salmon or Middle Fork Salmon are trips of a life time
and I hope you and your family get to float them.
It appears to me you have put some strong specifications on your trip. Gonna be hard to get the trip done on the cheap, my opinion only.
Having said that, if you are flying and going commercial with three people it is going to be really hard to do it much cheaper than 4 thousand bucks USA.
Only tip I can offer is for you to drive to Boundary Creek, get a shuttle driver to take you car down to Salmon and hitch a ride from the take out to Salmon ID. That means driving all the way which will take probably at least a couple maybe three days. You can run the numbers.
And, my bet is it will not ever get any cheaper the way things are going in the US.
Time to start saving big time and find a way for next season as commercial trips on these rivers often times fill up fast especially during the warmer float times.
I have floated both these rivers commercially and private trips. Both are exceptionally good times. Commercial floats are really deluxe with gear, guides and especially food being outstanding. Private may end up a bit cheaper unless you have to buy a lot of gear. But, you will do a lot more "work" on the trip. Advantages on either method. Commercial might sound like a lot more expense but when you figure in the fact you get to go the time you want to go, all you do is show up at the put in and all the logistics are taken care of including the expertise and gear needed to transport you down the river in style and safety - going commercial has a lot going for it on value for money spent.
I have used Salmon Air several times and they are great. The scenery from the air is outstanding on both Main and Middle Fork flights.
Good Luck and if you do make the trip, please give us Buzzards a trip report.