Keiler: I had or better still have this problem, but I can circumvent it by switching to the desktop as soon as the DOSbox launches in fullscreen (Alt + TAB or Windows Key + D), it now should run in a window and the game starts with graphics, you may now switch back to fullscreen (Alt + Enter) and the game runs fine. If it doesn't work, the game might have started before you got to the desktop.
good luck, have fun
If Dosbox doesn't have a problem with displaying LotR1's output in windowed mode, then it might make the whole thing easier by not having Dosbox start in fullscreen. Let it start in windowed mode and then hit Alt+Enter for fullscreen.
Open ..\LotRealm_RE\Lords1\dosboxLords1.conf with Notepad:
"fullscreen=true" - set to false for starting in a window.
But, who knows, there might be merit in playing with "output=OpenGL" to fix the whole matter more properly. You can replace OpenGL by surface, overlay, openglnb and ddraw.
One of those may allow Dosbox to show the glory of conquest in fullscreen from the beginning.
I can't test this, as I don't have that problem (using a Dosbox installation under Linux). Good luck.
Addendum:
For aesthetic reasons you may also want to take a look further down in the file. The "aspect" option helps, if you don't want the game streched over your whole screen, and with the "scaler" option you can choose from a number of filters that smooth out the pixels if you want that.