I agree there should be a counter-bad-draw mechanic, but I don't like mulligan. For decks relying on one card, as already noted with the lobotomizer example, mulligans can be abused to get that one card early. At least until there's more cards, that doesn't seem like a very good idea.
I'd rather see new mechanics that can be used for the whole game. For example:
Pillars only: when you posess no pillars, you can discard one from your hand to draw a card. This'd work only on your first turn, or when all your pillars are destroyed or stolen, obviously.
No pillars: each turn you do not play a card (including a pillar), you get two quanta from your mark instead of one. Again, works on your first turn or when you're locked and can't do anything.