Neither the inbuilt d3drm.dll or the native d3drm.dll will work however the d3drm.dll that came with my copy of LRR does work if put in the same directory as the game executable. You may also have this file on your game disc. It seems to be the version that comes with DirectX 6. I'm guessing it's a issue with newer versions of it that for some reason don't like Wine+LRR.
A simple way around the video issue is to delete or rename the two avi intro videos located in the data/AVI folder. This also means that it won't run if you have the game disc in your drive as it tries to run the intro video from off the disc. Obviously this means that non of the videos will play.
Another thing to note is that it only works in RGB Direct3D emulation mode when executing LegoRR.exe directly however it runs much quicker compared to it running in a XP VM on my machine. (A Core 2 Duo E7400 at 3GHz.)
To make sure your using the right version here's the md5 and sha1 of the working d3drm.dll.
MD5: 7e4541c04a1678c656875596db13599d D3DRM.DLL
SHA1: 77709ddd4963426c8c6755d160c31af35144faeb D3DRM.DLL