I have done this. I used a freight broker to book an independent trucker who owned his own flatbed. So I supervised loading (in my case, because of where the container was, and the weight (40,000 lbs plus or minus) I needed to load it with a crane. Then, the truck drove straight to my place, I flew up and met it there, and supervised unloading (again with a crane).
When loading the container, I bolted some things to the floor, some clips in the internal rails with straps, some crated, and some just wedged and stacked with no room to move. This was my container, that I had owned for some time before, and for another ten years after.
It all worked out fine.
The rate was a flat rate, not based on freight classification.
My broker said- a flat bed, hauling a container that weighs this much, is this much money. Take it or leave it.
The freight classification thing comes in when you are working with big LTL companies, usually.
At the same time, I got Schneider to drop off a 45 foot trailer for a week, which I loaded, then they came, picked it up, drove it 1200 miles to my new place, and left it for a week for me to unload. Cost me a bit extra for the parking time at each end, but it enabled me to carefully load everything and make sure it would not shift during the move.
This was some years ago, so actual prices are way out of date, but it was much cheaper than LTL, and much much cheaper than machinery movers or moving companies.