Try this and let BIAB do the transition for you by using the Song Medley Feature.
Tools| Song Form| Medley MakerTo make this work, you will need to save up to three versions of your project because Medley Maker will not start a Medley in 3/4 time So... for saved version 1
Select any RealStyle that's swing 8, 4/4 at 90 bpm and key signature A
Save As - Silver Bells_4-4
From the original version of Silver Bells that you posted on line,
Save As - Silver bells_A
The third song you'll likely have to experiment to make it sound to satisfaction because there are different ways to program the Medley Maker.
The options I see are:
Select the Silver Bells_A again and set your beginning bar at 53 and see if the Medley Maker plays the proper B
Listen to the song in both keys, A and B and compare the results
or
Save As Silver Bells_B after transposing the original Chord Chart to B and change Bars 53 forward without the modulation
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For the first song, select beginning bar at 1 for two bars and no transition
The second song is Silver Bells_A beginning at bar 1 for 51 bars and 1 transition
The third song is Silver Bells _B in the key of B beginning at bar 53 for 36 bars and no transition
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Once you generate the Medley Maker, it will create a new Chord Sheet composed of the Three songs as you've programed them. Bar 54 should be your 1 bar transition.
Edit the transition bar to the held F3...
In testing, I also edited the 4/4 first two Bars to having only an A. Chord and only the drums generated to make editing out those two bars easier.
I've included a screen shot of my Medley Maker settings.
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Sometimes you can achieve the desired results and have BIAB perform with different songs (or different versions and sections of the same song

) something BIAB can't accomplish within a single project. I hope this helps you get the results you're looking for even though you may have to experiment a bit. Good luck.