Actually, and I have argued this before, multiversal beings shouldn't be a problem for a prepped Doctor. The Reality Bomb and the Time Lord's Ascension both are capable of destroying the whole of reality; which would include Lucifer. In addition to that; the Doctor can absorb the Time Vortex and become the Bad Wolf which has complete control over all space-time and can see all that ever was, is, and ever could be. In addition to that the Time Vortex exists everywhere, including in other universes as well as the hub to all realities (Medusa Cascade) making it omniversal. Having Omniversal control over all space-time (to the point where you can create paradoxes and create yourself, as well as completely undo death forever) and Omniversal Omniscience should put him ahead of Lucifer quite easily. And that takes very little prep (just pop open the heart of the Tardis). He doesn't do it because it would make him a vengeful god; but he easily could do it and he has done it in the comics to save people.
Time Vortex doesn't apply if you are beyond time. In the Whoverse, beyond the Fifth Dimension is ruled over by beings such as the Guardians, who are above the universe destroying Chronovores, etc. While singular Chronovores or Eternals can be destroyed, as a whole nothing in reality can stand up to them. Lucifer is beyond space/time. He has the capacity to manipulate anything coming at him, he is the compliment to Michael's infinite creation. IIRC, Lucifer could step outside of reality. That alone means if he doesn't want the Bad Wolf Entity to harm him, it can't. Other beings in DW, such as the Lampreys and the Memeovore, are acausal. Changes in time, attacking them with time, etc. have no effect on them. Theoretically, enough temporal energy should be able to kill anything, even the Elder Gods and the Guardians, but the sheer amount needed to would be astronomical. The Doctor burned up thousands of universes worth of temporal energy just to "kill" the Lamprey, and even then it had effortlessly eaten an infinite multiverse down to a set size.
The Time Vortex itself is nothing next to the Quantum Archangel, which is all energy in existence wielded by a single being. It was handwaving billions of complete universes into existence, and that was with the thought process and senses and awareness of a human. Heck, the Time Lords could snap the Vortex out of existence if they wanted to (Eye of Harmony), and thus the Daleks could to. The point being, while powerful within 4 dimensions, the Vortex is less useful beyond that except as a glass canon, and even then the sheer amount of energy needed to kill cosmics in DW dwarfs anything any Time Lord or less than 6 dimensional being is going to be able to bring to bear.
So effectively, Lucifer stomps on almost everything. He has infinite manipulation of things, and thus can handwave away any attack that the Doctor could send at him short of the Doctor having the Quantum Archangel powers. (sidenote: not a Lucifer reader or anything, that is just what I have heard/seen on here and other sites
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