Giving the “coup de bec” (how do you say this in English?) being quick, it is in the “passé simple” (historic past).

On the other hand, taking off the burden from the animal lasts a bit, and is therefore in the “imparfait” (imperfect).

This is the usual distinction for “passé simple” and “imparfait”.

Similar sentence and usage in English, with past continuous instead of “imparfait”, and preterit instead of “passé simple” :
She gave him a quick kiss while he was taking off his raincoat.

In effect, when reading French literature that uses the passé simple:

• The passé composé is replaced by the passé simple, except in dialogue and similar text intended to directly represent speech. (I’m quite sure that Harry and the other characters do not use the passé simple in their speech.)
• The imparfait remains the imparfait. In all cases in which you would use the imparfait in spoken French, it shows up in written French.

In other words, the general difference between the passé composé and the imparfait that you learn remains, except that the passé simple is used instead.

In your example, the contemporary natural spoken way to say that sentence would be:

Elle a donné un affectueux coup de bec à Harry tandis qu’il lui enlevait son fardeau, puis elle a traversé la pièce d’un coup d’aile pour rejoindre Errol.

• Donna/a donné represents a completed action (she gave him a peck and it was done)
• Enlevait is in the imparfait because Harry “was taking off” the burden, an action that is ongoing. You would use the imparfait in spoken French in the same way.
• Traversa/a traversé also represents a completed action (she crossed the room).