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.

Not as good: "He came over..." emphasizes the fact that he came over. This is not good because it's not the main topic (what he did after he got here).

Good: "He had come over..." implies that we already know he did come over and leaves the emphasis on the new information (to do a single) where it belongs. However, if you also didn't know whether he came over at all, then "He came over to release a single." would be best.

Shifting emphasis (based on textual clues only):

"He came over to release a single." (two competing emphases)

"He had come over to release a single."

By the way, you can say either version, most people won't notice the difference. But, the more you want to emphasize what came afterwards, the more you will want the "had come" version. Example:

Bad: "He came over to release a single, when his mother died."

Better: "He had come over to release a single, when his mother died."

I note that in your example you are specifying that he said "I came over" (past) rather than "I have come over" (present perfect). When they backshift, both of these shift to "he had come over" (past perfect).

In your example, the version "he said he had come over to release a single" reports the tense he used to describe coming over in an unmarked way, and the version "he said he came over to release a single" describes it in a marked way. It sounds as though you don't want to mark it. But one reason you might want to mark it would be if you wanted to convey that he himself wanted to emphasise that his coming over was in the past and that it no longer had continuing relevance, or that it no longer had its expected continuing relevance. So for example the second form works well in this altered version:

I met him last summer. I asked him what he was doing over here and he said he came over to release a single but then unexpected things happened and he was about to return home early.

Try emphasising the words "came over" when you read that. This version is more oral, it takes you to a greater extent to his time-frame, or in other words it puts more of his frame and therefore more of his point of view into the mixture.

This is not to say that you could not also use a backshift:

I met him last summer. I asked him what he was doing over here and he said he had come over to release a single but then unexpected things happened and he was about to return home early.

Yes, it's because his coming over was prior to anything he "was doing here" which is what was asked of him.

The passage contains reported speech, which needs to be bacckshifted to reflect the frame of reference of the time it was spoken.

So "he'd come over" correctly reflects that the coming over was prior to the (past) reported speech.

In the absence of the direct speech of reference, we can shape this sentence either way, simple past or past perfect.

I have certain reservation about use of " here" in the passage. It reads fine to use " there", and rule books also approve of the same. But there is a strong reason to retain ' here ' as what follows would , in that case, be difficulty to locate in place. I duly acknowledge this suggestion of retaining 'here' based on sound logic hereinbelow in the comments.

Perhaps, the dirct speech was something like this:

He said, "He has come...".

This present perfect has its corresponding past in simple past tense or past perfect tense abiding by the rule of back shifting. However if something is to be distanced further in time, better use past perfect tense. There is nothing wrong in using simple past if the direct speech in in present perfect.