So you want your keyboard to continue playing its current notes with its current sounds while you change programs, then use the new program sounds when you release the keys and play again? That would be a function of the keyboard, and I don't know if any keyboard does that, but you could ask the manufacturer.
If you wanted to stagger the program changes so they're called at different times, you would have to move the program changes for the two keyboards into two separate MIDI presets and attach them both to your song. Then you could trigger one with a song selection and trigger the second one with a foot switch or a button on one of your keyboards or an automation track. If the timing is consistent, you could create an automation track that triggers the first preset when you select the song, then triggers the second preset after a specific delay. I'm concerned this wouldn't be practical if the timing for each song will change depending on what song precedes it. But since you only want to do this for a few songs, maybe that will work.