The argument that congressional chaplains are constitutional is not that they have to be Deists but that it has been an old tradition (predating the Constitution by 16 years IIANM) and that their function is mainly ceremonial.
Even Madison (as well as other founding fathers) recognized that this practice wasn't kosher according to the constitution but he concluded that is was de minimis and thus he decided not to rock the boat and give it a pass since there were more important matters to be addressed at that time.
Of course if congressional chaplains didn't exists and someone wanted to introduce them today then this would most likely be ruled to be unconstitutional but since it's a long-running tradition it's deemed OK.
Even Madison (as well as other founding fathers) recognized that this practice wasn't kosher according to the constitution
I looked this up. Madison only expressed this view long after he retired, in the so-called "detached memoranda." This was a flip-flop, as he was on record as supporting the chaplaincy while still active in politics. In addition, I could not find another founder who who agreed with him.
The clincher for me is that George Mason never once objected to any Federal public displays or references to religion, which were a commonplace in those days.