The emergency engine could be massively less sophisticated because of its rare and very short term utilsation - like a cruise missile jet. It could be cheap, noisy, polluting, dreadfully inefficient but above light
You mean a pulse jet?
My questions would be: what do you do in an EFATO? You'd need an engine that can be started reliably and almost instantly. Hobby pulse-jets don't fall into this category, but perhaps commercial aircraft could do better.
Secondly, if it's less efficient than the usual engine, you'll need to carry significantly greater fuel reserves which might well negate the savings from not carrying the initial engine.
e.g. an A320 engine weighs about 2 tons. For the sake of argument your pulse jet weighs 1 ton. The fuel weighs about 20 tons. It wouldn't have to be much less efficient than the original engine to totally negate the weight savings. It might be less costly to maintain, I grant you.