Yes some tow companies do these kinds of hauls all the time. Not all do though. When I did it for a living, the company I worked for did it. There were only 2 of us, out of about 30 that were set up to do it. Part of it is having the right truck, but most of it is having the right driver. Not all equipment can be tied down in exactly the same way, but it all must be properly tied down. This is too much math for some drivers, especially if all they know how to do is tow a car. But it's not a problem at all if you get a driver that knows what he's doing.
When I did it, equipment hauling was charged by the hour, and not by the mile. The clock started when the truck left the yard (or last drop off), and the clock didn't stop until the truck was back at the yard (or next pick up). There was a rate for light/small equipment, and a rate for large/heavy equipment. Yours would have qualified for "light/small" equipment.
U-haul *may* have a trailer that will work for you. It's been a long time since I used a U-haul car trailer, but seems to me theirs are not true full flat bed trailers, but only have channels for the tires of the vehicle to go up the bed (can't think of the correct term at the moment, not enough coffee). If this is the type of trailer U-haul offers you to use, just measure the tire track width on your tractor to make sure the tractor will fit the channel width on the trailer.
A full width flatbed trailer won't have this issue.
A standard car trailer will haul that load (3200 lbs) just fine. If you have a buddy with one you can borrow, or a local rental place rents actual "car trailers", that may be the way to go. Don't forget chains and binders, one on each corner, to tie the whole thing down with.