I drove from Vermont to California and back with my DDs (8 and 12 at the time) last summer. The trip was a month including various stops.
We had an awesome time. We joined AAA and called ahead, sometimes the same day or sometimes 4-5 days ahead, to motels in their guidebooks. For a place like the Grand Canyon you need to make reservations well in advance. We ate out 1 meal per day. We had a cooler & stopped at grocery stores for cheese, fruit, bagels, etc. for the other meals; also sometimes breakfast is included at the hotel.
I found that with stops I could average about 50 miles per hour over the course of the day. Therefore to go 2270 miles, it would be 45 hours, or 5 days of 9 hour drives, for example. You might be able to go further in a day because you have 2 drivers, but you have a younger child so might need to make more stops. Also, places you want to stop are not necessarily a full day's drive apart. If you do a short day you will either have to compensate with an extra long day or you are going to end up driving more than 5 days.
Personally I would not do it with a 4 year old. DH and I drove from FL to VT when DDs were 6 and 2 and we decided to drive through the night so the kids could sleep and not have to be just sitting in the car.
Also renting a minivan is expensive and I don't know why that is necessary if there are just 3 of you.