A lot of it depends on the personality of your child. How is he with long car rides? I've found in my own personal experience that's a pretty good indication, but every child is different. My oldest son is currently 11, high functioning autistic, and deals with ADHD. His brother, who is 7 is a typical energetic child. Who is the easier traveler - my 11 year old! Like me, he loves staring out the window and watching the world go by. Our 7-year old gets bored with it quickly and he's the one we need to keep busy and make sure he's sleeping well and taking naps. Our trip from St. Paul to Flagstaff last year was a hybrid of coach and sleeper. While my oldest was ready to take the trip for several years, we waited for a couple extra so our youngest could handle it. Again, just personalities at play. I've never seen anyone attempt using a car seat in a coach seat. I've just watched them bring them on and stow them. Your child might like being able to better see out the window if you use one.
If your child likes to travel, that will be a big help for you. If you happen to deal with a meltdown, there are bathrooms in the lower level you can take him to if you do need a little privacy to get things calmed down.
And while I have no desire to get involved with the coach vs. sleeper bantering that this topic seems to have degraded into, one poster made a valid point to check your costs. If you plan on going to the dining car, or even the lounge car for every meal, it will add up very quickly - especially for 2 adults and one child. You MIGHT come out paying for a sleeper and getting the meals included. Then again if it a peak time, you might just be paying a lot of extra money you don't wish to spend. Either way, it might not hurt to research it.
Also, you MAY be able to upgrade during the trip if you change your mind. You can check with the conductor and if you have a willing one, you might be able to score a room at the lowest bucket. Again, just throwing that out there as something to keep in as an option. I personally have not attempted it, but I know others on this board have. In my travels, I have seen a lot of young ones sleeping overnight in coach with parents and overall, they seem to handle it pretty well. Of course, there are always exceptions.
If you decide to make the trip, have a great time and I'm sure your little boy will never forget it! Be sure to let us know how it went.
Dan