Keep your plans flexible and keep track of conditions - it's always possible that an early season storm could close the GTSR before September 18, but it will definitely not be kept open later due to the reconstruction schedule. Technically, September 18 is the last day it will be completely open.

Just thought of something else - you probably can't take your RV across the GTSR - it has a 21 foot length limit, 8 foot width limit, and 10 foot height limit. However, there is a free shuttle bus system.