Four years ago we took our 24' Class C from Tucson to Maine stopping just about everywhere along the way. We left May first and returned July first. We had close encounters with tornados at least 6 times. Granted it was a very active year but they don't call it tornado alley for nothing.
The internet via your smartphone is your best friend and it enabled us to monitor developing weather conditions and reroute, proactively dodging a number of storms. I was more concerned with potential hail damage than an actual tornado as the thunderstorms associated with tornados are not to be taken lightly.
We saw damage in Arkansas from a tornado that hit a few days before we passed through and it was incredible to see up close and personal. Passing through Pratt Kansas on the return trip got more than a little exciting as well. Spent 24 hours hiding out from storms in Elkhart, the nice folks at Nexus asked us to stay an additional day for free at their factory facility with hookups to avoid an incoming front. The next day we hunkered down for another 24 hours at an RV park in the cornfields of Illinois where the two nearest shelters, farm houses a half mile away were on standby for our evacuation.
The OP's trip is certainly doable but lots of planning and caution are called for.
Never hesitate to change plans, routes, and destinations at the first sign of a serious front moving through. Always know where the nearest shelter is located and be prepared to bail in the middle of the night if the worst occurs. Staying hooked up and ready to roll on short notice is not a bad strategy if activity if in the forecast. The good news is that they tend to move through an area quickly and you will usually have an alternate route that can minimize any potential exposure. Good luck!
:E
