Every time I make a rash decision like the one you're about to make, I'm sorry for it every time I wear the garment. If you purchased a coat without interfacing you'd likely say it wasn't made well. I'm guessing you want your coat to be "well made"!
Elizabeth, I didn't interface my coat, not the current one (which I have) but another one, and I regretted it as it never looked good. It was quite a drapey wool I used and I could have easily used a fine interfacing (like Pro Sheer Elegance) and fused every piece and it would not have made a difference to the drape of the fabric but it would have given it some structure. Live and learn. However, my current coat is fully interfaced with the same interfacing as I commented above and it is great. My fabric is a drapey wool/cashmere, it took forever to fuse the interfacing but it was worth it, IMO. Perhaps you could do a sample, test the interfacing on a small piece of wool and decide if you like the feel of it. If you use a fine enough interfacing it should not affect the wool or make it feel stiff.
I say interface as directed. It sounds and looks like a classic tailored coat to me and if you are using a nice wool or something you will want to give it body and structure not to mention a lovely finish. But - then again, I am a rule follower! Have you used the book called "The Classic Guide to Tailoring" The Classic Guide to Tailoring the Perfect Jacket -you can get it on Amazon. It is so comprhensive and yet to the point. You can't really cut corners in tailoring - just my opinion! Happy sewing!