You can install a newer DOSBox over the top of the one included with the GOG installer.
Or alternatively, use the GRP files from the GOG install with eDuke32 for a native Windows experience.
All I have to do is install the newer version of dosbox over at: C:\ProgramFiles\GOG.com\Duke3D. Right?
As far as I know, yes, you can do that. The new DOSBox will then replace the old one provided from GOG's installation. However, I very much recommend using eDuke32 instead. You can still play the game as it was with 2D sprites and classic controls, or you can up the resolution, add 3D models, high res textures, etc, and a modern control system.