I see your choice fundamentally not as whether or not to backshift, but whether or not to use a perfect tense.
The present perfect tense carries a connotation that is not present in the corresponding simple past, namely that the corresponding action is recent and/or relevant (that is, it continues to affect the present). "I have cleaned the room" suggests that the room is still clean, while "I cleaned the room" could mean that you cleaned it far enough in the past that it has had time to get dirty again. You can add a time expression to clarify when an action in the simple past occurred ("I cleaned the room yesterday"). By contrast, unless you want to express something like repetition ("I have cleaned the room many times over the course of my life"), you're theoretically not allowed to add a time expression to the present perfect (though this rule is often broken, especially in speech) because it already inherently suggests that the action is still relevant.
Using the past perfect ("I had cleaned the room") can be used to indicate ordering, as you suggested ("I had cleaned the room when she walked in"). However, it can also be used to show recency/relevance, the same way the present perfect does. The present perfect would be used in the actual dialogue ("I have come over to release a single") to indicate a degree of recency/relevance that would not be present in "I came over to release a single", because the speaker could theoretically have completed the sentence with "... last year, and then I went back."
In reality, context would make it clear that he came over in order to release a single and was still there, so the action of coming over was still relevant. Thus, no one would be puzzled by "he said he came over to release a single". But using the past perfect makes the relevance more clear.