The Biddle Trick has been a go to trick for me for years. The justification for splitting the cards is because you're trying to cut to their card, but I see the subtle point your trying to make. Why actually cut the top half to the table or into their hands? Why not a simple in the hands cut? I've thought about this before, and while I still perform the original version of the Biddle trick a lot, I've come up with a few solutions to this problem in my years of thinking about this one trick. I'll won't really go over methodology much, because I want you to do your own research behind a few of the moves I'll discuss. Here's a couple to play with.
Solution 1: Control their selection second from the top via your favorite method. Say, "Now I've spent years and years handling cards, honestly, I could find it. However, what if I put you to the task of finding your own card by cutting to it? How do you think you would do?" A little by play and you say, "Actually believe it or not, it's not that hard. The trick is not even thinking about it. I'll tell you what why don't you cut about half the cards over there, and I'll mark where you cut." During the preceding patter all you've done is the Max Holden Crisscross Force. Look at the spectator and ask, "So how do you think you did?" The force, followed by the direct question, followed by the spectators answer and a little by play will build up time delay. Just enough to make the force go by. Proceed as per the original handling.
In that variation the spectator does the work, taking the burden off of you. As a result, the audience won't think about it, nor can they reconstruct how the card ends up in the middle; because you gave the spectator complete control.
Solution 2: Get Jon Racherbaumer's book on the Devilish Miracle or Card Finesse II. There is a really good Marlo move in those. With that one move you could very well satisfy the criteria of up-jogging cards versus cutting the deck and still load the card into the middle. I do this sometimes for a very direct handling of the Biddle Trick.
Hope this inspires you to come up with new ways to do this wonderful trick.