Anything is possible, I think you will get caught up in some issues.
- Typical RV rental places likely won't have anything like this. It will have to be a specialty company.
- The driver will need a commercial license (generally RV's in the USA do not require a special license regardless of size)
- It would have to be a motorhome of some sort as several states don't allow riders in trailers and even those that do have rules.
- Once you get to your destination how will you get around? We don't have transit like you expect in Europe and a big motorhome isn't convient for day trips (not to mention your driver will already be dead tired).
The kids may be the ultimate stumbling block. Legally, adults do not require seatbelts unless in the front seats. Kids must be in seat belts regardless of where they sit. Realistically most kids in motorhomes aren't in seatbelts...but they are typically driven by relatives who take a calculated risk that they won't be caught. The problem is a professional driver who's license is on the line and the company that may have seperate liability concerns may not want to deal with the issue.
With only 10-14 days, I would set more realistic goals. Imagine if someone said they wanted to see the UK, france, germany and italy in 10 days. While it's theoretically possible, it won't be a fun trip.
Another option would be to consider a rail trip with stops at hotels and rental cars mixed in. Not neccessarily cheap but solves a lot of the complications.
