Most likely, the hitpael form in the case of התפלל is doing the same thing as the hitpael form in the case of התחנן: it is turning the word into a voluntary passive/indirect reflexive. Hence, the meaning of התפלל is to make oneself the object of God’s פלל (assessment, intervention, or judging). This is a much simpler understanding of התפלל than the ones that look for a reflexive action on the petitioner’s part.
Mitchell First, “The Meaning of the Word Hitpallel (התפלל)”, The Seforim Blog (29 August 2016) [http://seforim.blogspot.com/2016/08/the-meaning-of-word-hitpallel.html]