That's a big space and a beautiful building. I've heard of it and even seen old woodcuts of the exterior in books about the American Civil War but never seen the interior until now. A very nice example of its period and location.
Looking at it, unless your congregation can fit in the front three or four rows I'd be inclined to use subtle sound reinforcement for a small choir in there even without competition from the organ.
Pipe organs can sound amazing, but the volume that comes out of them at working air pressure is the only volume setting there is of course. Maybe encourage the organist to pull out a few fewer stops?
Fwrend is absolutely right that getting things balanced sound-wise has to be a process of taking everyone with you if it's going to work. In the end it's a win-win game because everyone benefits - musicians, singers, congregation, leaders and speakers because everything can be heard, no one is deafened and no-one has to strain their voice any more.
The main problem I can immediately imagine with the single mic at several feet scenario is that to get the mic's gain high enough for it to be doing much you might be getting much closer to the brink of feedback than you'd like. Spill from the organ (not just direct but from the building's natural reverb which will also be time-lagged) might be a problem as well. The further away a cardoid mic is the more spill it"s going to pick up and re-amplify and we all know where that ends up.
On the other hand, obtrusive mic stands would look out of place in that setting and I doubt you really need a mic for every singer anyway, and sound-checking would become a significant exercise if you did. There are mics and stands that are intended to blend in or "disappear" such as the AKG DAM+ range. The catch is that many of them are intended to be stood on a desk or lectern so are only 18" or so long. It might be worth your while having a chat with your local theatres, who might face similar problems of balance and projection, and find out what they are using. If individual mics are an acceptable solution might headset or lapel mics be a possibility?
The other thing to consider is speaker placement (how many as well as where), unless you've already got an adequate setup.