I spent two hours trying to get it running. This thread was the first result in google but had no solutions. This is how I got it to run, I hope it helps someone in the future.
1) Install the game with the GOG installer
2) In the installation folder, locate and make a copy of both files in VtMB/Unofficial_Patch/dlls/ (vampire.dll and vampire.dll.12)
3) Optionally, copy the Unofficial_Patch folder and rename it Companion_Mod
4) Extract the vampire folder from compmod_1.0.zip into either the Unofficial_Patch or your Companion_Mod folder (you don't need the other files from the companion mod)
5) Extract the vampire folder from compmodv1.4Patch.zip into either the Unofficial_Patch or your Companion_Mod folder
6) Overwrite the files in the dlls folder with the ones you copied earlier (THIS IS THE CRUCIAL STEP)
7) Right click on vampire.exe -> Properties -> Compatibility _> check "Run this program as an administrator"
8) If you created a Companion_Mod folder, edit the properties of the VTM Bloodlines shortcut and change in Target the "Unofficial_Patch" to "Companion_Mod" (or whatever you called your folder)
9) Launch the game through the shortcut
This worked for me and I hope it works for you.
tl;dr: You need the vampire.dlls from a recent Unofficial Patch to get companion mod to run on recent systems.
Not sure what else to do.