Even though Logic has flex pitch and flex time which can both be very useful I prefer actually transposing entire sections of audio files rather than use the transients that the program analyzes for something like what you want to do.
First make a duplicate of the audio file for safety because whatever you do can usually be undone but it is destructive. Select the audio file in the main window and under the main “window” menu select “open audio file editor” (or command-6). Here you can select the segment you want to transpose or the entire file by click dragging. Under the “functions” menu select “time and pitch machine” to open that window. You should be in free mode, not classic, and the destination tempo should be 0% or the same tempo as the original. The transposition is at the bottom in cents so an octave is 1200 cents. Negative values transpose down, positive, up. You can preview the sound (sometimes it previews at a different tempo, don’t know why) or just hit “process and paste”. You can still undo at this point if necessary.
One thing to be aware of, if you transpose any audio file a whole octave you are likely to get a very unnatural sound and also possibly a lot of artifacts. I suggest singing your part in as low a key as you possibly can so your transposition value is not as extreme. For example if the song is in A and you can sing the part in C then you only have to go down -300. Good luck, hope this works for you.