And with this we can understand that which is written, "You made a covenant with him", and Rashi brings from a Midrash that Hashem holds on to him. And if such a thing is written in the Torah, then this matter is eternal, and as such, at every bris milah Hashem holds (as it were) with the mohel. Because just like a seal that is on a document cannot be done through someone else, with another person signing this name, this would be regarded as a forgery, and every seal is not worth anything unless the the owner (of the letter/document) himself seals it, so too with a bris milah, Hashem Himself seals it and testifies on the bris (the covenant) between Himself and Israel.
And with this we can now answer the question in the Teshuvas Chasam Sofer why we call the Sandek the 'baal habris' (lit. the owner of the covenant) even though the main mitzvah is done by the mohel. Since we can undestand that mohel is to be regarded as only like the quill that is held in the hand of Hashem's seal Himself. But the main bris is performed by Hashem Himself and that is how we understand "You made a covenant with him".
I appreciate that I am missing the rest of this piece, and therefore I am missing some points that I would love to have clarity on. (If anyone has a link to the full shtickel Torah I would love to see it in its entirety).
This only highlights why we don't refer to the Mohel as the 'baal habris' but it doesn't explain why the Sandek receives this term?
Does anyone know the source of the Chasam Sofer?