I have pulled through 40 of the fifth states in the US and have never ever had to enter a DOT road scale. I have even had a stations turn on a LED sign saying you do not have to enter the scale house. This blew my mind the first time I saw this in WI on I90 near Janesville after you enter the state.
I have written different States DOT when I first had my first 5er back in 2007. And all of their reply's where if you are a private hauler towing a non-commercial trailer then you do not need to enter a scale house.
My total weight of truck and trailer is slightly under 21,000 LBS when towing.
What I would worry more about are you pulling with the correct driver licenses? Most states have a limit of less than 26,001 LBS on a standard drivers Licenses and anything over that you need to have a different licenses. Read your Drivers Licenses and see what it states.
2016 34RL CC; 2008 Ram Mega Cab 2500HD, 6.7L, 68RFE 6 speed, 4X4, Smarty S67