The correct option is
C In case you are not happy with my work, please let me know.
Conditional conjunctions are subordinating conjunctions which are used to connect two sentences when one sentence describes something that did/ will happen if the condition of the other sentence is satisfied. Example of conditional conjunctions are: If, unless, in case provided, in order, as soon as, etc.
It may be observed from the two sentences, that the second sentence is dependent on the first. "Please let me know" is conditional upon, "You are not happy with my work". "In case" is the most appropriate conditional conjunction for the given example. It means the same as: "If you are not happy with my work, please let me know". Hence, this is the most appropriate conjunction to be used. Option C is correct.
Option A: "On the contrary" is a conjunctive adverb which connects two sentences which express contrast. Here, the two sentences do not contrast each other. One is conditional upon another. Hence, this option is incorrect.
Option B: "As soon as" is also a conditional conjunction. However, it connects two actions/events, which happen one after the other, almost immediately. i.e at the very moment one action/event is completed, the other one happens. In the given example, the two sentences are not related by time instantaneously. Here, one sentence is conditional upon another. Hence, option B is incorrect.
Option D: "Not only... but also" is a correlative conjunction which is used to express two clauses which belong to the same parts of speech. If the word which follows "not only" is a verb, then the word which follows "but also" should also be a verb. Here, "not only" is followed by a pronoun (you). "But also" is followed by "please", which is an adverb used to emphasize politeness. "Not only.. but also" is used to connect two sentences which complement each other. Here, one sentence is conditional upon another. Hence, this option is incorrect.