Cool concept! I'm not sure if this is just me, but the shellinabox seems quite laggy in handling key presses.
One technical point though: if you are intending to set up DoomRL to run in a browser, then you need to channel sounds over also because sounds are important in the DoomRL universe -- it basically provides important hints on where the enemies are that are not visible, particularly since DoomRL is more about ranged combat than melee like nethack or rogue. I'm not sure if shellinabox is able to do that, but you might need to deal with that eventually.
Meanwhile, I suggest trying to get DiabloRL online first. I believe the restrictions are less so compared to DoomRL, and sound is not that critical for game play.