For those that don't live in the Illinois Midwest area....
May is VERY unpredictable in Illinois, been there done that. :B
The month of May is not an end all be all for what is called Spring in Illinois. Actually very 'few' Memorial Day weekends have weather that is ever a pleasure to be camping.
That said. The batteries will go dead before sunrise and the furnace in a TT is a HOG on propane. Do you have one or two tanks on your TT??
Many RV'ers do use the MR. Buddy Heaters, propane heaters designed/rated to be safely used indoors.
This small one is ok but I worry about the tip over factor from the small base. Used with caution away from kids and pets it should work for you.
This is the most popular model for RV's
Found at any big box camping store. Gander Mountain, Dicks, Bass Pro etc and Tractor Supply.
http://www.cabelas.com/catalog/product.jsp?productId=715121&destination=%2Fcatalog%2Fproduct.jsp%3FproductId%3D745033%26type%3Dproducthttp://www.cabelas.com/product/Mr-Heater-Portable-Big-Buddy-Heater/745033.utsThey both run off those small propane bottles used for camping stoves. IMHO I would still always leave a window open when using it inside an RV.
Good Luck. Happy Camping! And here's hoping for a good weather Memorial Day weekend for you! :C