Put short: we do not know how to interpret had brushed my teeth. Which reference point in the narrative does it refer to? To the one just mentioned (took a shower) or to the earlier one (I realized). Unless you make that explicit, the sudden had brushed my teeth is intrusive and awkward and impossible to interpret with 100% accuracy.
Keep in mind a few things.
The past simple is used to express each successive event in the past narrative. The reader will presume that each new event follows after the previously mentioned one (unless told otherwise). This is why the suspect's first statement is easy for the reader to interpret. You have told us this is a chronology, but that is what we would have interpreted it to be even if you had not told us.
Next, each event in the past narrative (which, again, are marked by the independent clauses using the past simple) also sets or establishes a reference time or point. And any past perfect will be attached or tethered to a particular reference point by the reader, if there is enough clues in the text to allow her or him to do this. If there are not, then it is bad luck for the reader.
The insertion of the past perfect sentence I had brushed my teeth is unnatural. This is because the previous past perfect I had woken up a little late not only indicates that wake up occurs before I realized... (which so far is the only event in the narrative). I had woken up also prepares the reader to return to the chronology of events, with each consecutive action expressed by the past simple. The sentence I took a shower is fine, because it continues the narrative. Note, the reader will presume that each new event in the past simple takes place after the previous one in the past simple, just as in statement 1. Note also that each added event (which will be expressed by the past simple) establishes a new reference point in the narrative.
At this point you come in with another past perfect I had brushed my teeth. This is sitting there and we do not know how to interpret it. It is not tethered or attached to any previously stated event or reference point in the chronology. Since it follows after I took a shower, I as the reader try to infer that this is the reference point you are talking about that had brushed my teeth is supposed to refer to & thus "come before," but I cannot be sure of that. There is no clear signal to me that that is how I am supposed to interpret it. It is sort of a free-floating past perfect that is not tethered or attached to any specific past reference time. This makes this sentence I had brushed my teeth extremely unnatural and awkward here.
Being a past perfect, it is meant to be interpreted as taking place before some reference point. The problem is that there is now more than one reference point in the chronology. There is I realized... and I took a shower. As a reader I interpret past perfects according to given reference points. Since there's more than one reference point, I have no idea which one you are talking about--grammatically speaking.
To help the reader correctly interpret your intended use of had brushed my teeth, you need to make it explicit which reference point it is supposed to go with. You can do this by writing:
Then before I took a shower, I had brushed my teeth.
The then is useful, because it transitions this new reference point into the sequence of past events. It is also now clear to the reader which reference point you are referring to with this new past perfect had brushed. And at this point, the narration continues, with each newly-stated past event creating a new reference point.
