“How could loving the deer too much possibly be the mistake which lead to Bharata’s rebirth in the body of an animal?”

First, we have to stop calling Bharata’s action a “mistake.”

Śuka never describes it as a mistake, he praises Bharata.

The only slightly critical word Śuka ever uses regarding Bharata is that he went “very far” in his affection for the deer. Further then duty required.

This is still not a “mistake.”

Bharata’s experience with the deer appears to be a mistake, flaw and disturbance to the early practice of his yoga, but in fact it is the cause of his yoga achieving the heights it actually does in the end.

So I think the best way to understand the story is to stop believing the common misconception that Bharata made a “mistake” and his birth as a deer was a “punishment.” The fact is that Bharata acted in an exemplary manner, as always, and his rebirths were part of the path to award him with much higher realizations of his ultimate goal.