Tennessee has reciprocal agreements with Arkansas, Florida, Georgia, Louisana, Kentucky, Mississippi, South Carolina, Texas and Virginia. There are a number of other states that may honor your permit based on their state law. For instance, Idaho, Indiana, Michigan, Montana, Vermont and Wyoming honor TN. carry permits but their state permits are not recognized in TN.
As a certified Police Firearms instructor in TN., I would think that a motorhome would be considered a moving vehicle. If you have a valid permit from S.C. and you enter TN., you would be OK due to the reciprocity. Many states require that you have the weapon unloaded, cased and not within reach of the driver i.e. in the trunk or locked someplace in the vehicle if you do not have a permit. Some states such as Mass. or N.Y. will confiscate the weapon if found in your possession even if cased. I would look at the states you plan on traveling through and contact the Atty. General's Office. If you don't want to bring attention to yourself, you might try some of the firearms associations in each state. The Tennessee Firearms Association, Inc. web site is
www.nashville.com/~tfa and their email is
tfa@nashville.com. They might be able to help you. Don't forget, the NRA site is also there and may be able to help you.