This custom is recorded in a number of places. The Artscroll book on Bris Milah elaborates on this subject in the notes on pages 110-111, and Yossele Weisberg Z'L also quotes a number of sources to this effect in the first volume of his 4-volume "Otzar Habris", in footnote 22 on page 181.
This is the reason why just about every siddur makes a space in the middle of a verse (which should really not be done) to indicate that this is when the recitation for a bris begins. It is because this recitation was included in the siddur that we now - virtually everywhere - distinguish between "Vay'varekh David" and "V'charos Imo HaBris." Even though this break should only take place at a minyan where participants in a bris taking place that day are in attendance - the demarcation was made, and a 'minhag ta'us' (mistaken custom) is the run of the mill in just about every shul I have ever attended.
May all of us be blessed to achieve a greater appreciation for the bris, its significance, and the role it plays in our lives. And if a tiny inconvenience in davening can help us at least on the morning of a bris, I think it's worth it.