You could send MIDI on song selection, then start the MIDI beat clock (start the tempo) on song second selection. That's a fairly common approach for doing things like starting a backing track, when you want a moment to look at the song before starting.
If you want it more automated, you could record an automation track for each song that sends the MIDI, then starts the tempo a moment later, then start the automation track from the song selection. You can set up one automation track, then copy and paste it to the other songs, but that would still be somewhat tedious to set up.