Go to Xbox live marketplace, and go to Doom or Doom 2. Does it say that you have to buy it, or does it say you already own it because you own the BFG edition?
If it says you already own it because of the BFG edition, I think there is a way to circumvent the idea that the disc spins while playing Doom and Doom 2.
Now me, I have the PS3 version of the BFG edition, and I already own XBLA Doom and Doom 2, so I couldn't really check for myself.