Exporting MIDI Files for Customized Sections

I'm currently working on a Karaoke App with a friend. My role entails creating MIDI files using Logic Pro, while my friend is responsible for converting the MIDI files to JSON format and linking them to Apple Music. Since Apple Music's audio cannot be modified, my MIDI files need to synchronize with the audio tracks provided by Apple Music. However, most Apple Music tracks have a few seconds of silence before the song begins, and when I import them into Logic Pro, I must align the song with the project tempo to ensure synchronization. This has led to the following issue:

Using the song "Back to December" as an example, I align the first beat of the audio file with the first beat of the second measure in the metronome. Consequently, the starting position of the audio file is at 1 3 2 152. I hope that when exporting the MIDI file, it can also start at 1 3 2 152 to avoid any misalignment when importing into Apple Music. Despite trying various export methods and even attempting to mark the starting position using Sustain, I haven't been able to overcome this challenge. The exported MIDI files still start at the default position of 1 1 1 1 according to the project.

Could you please suggest a solution to resolve this issue? I would greatly appreciate any assistance you can provide. Thank you in advance for your help!

(Logic Pro 10.7.7 / MacBook Pro Ventura 13.3.1a )