Excellent detective work, Jim!
This points out just how very, very hard it can be to track down noises on a bicycle. The tubes telegraph the noise so it is very difficult to determine the source.
As for the Brooks, the usual sources (by frequency of reports) are...
a) where the rails are riveted to the cantle plate at the rear of the saddle (or where the springs mount/are seated)
...or...
b) where the nosepiece wraps around the rail "return" at the front. In either case, inverting the bike and carefully and sparingly applying a single drop of oil to each of those contact points will tend to quieten the squeak/click without migrating to the leather of the saddle.
Another possible source (among many!) is the seatpost-to-saddle rail clamp or even the seatpost clamp at the top of the seat tube.
Best,
Dan. (...who thinks you did well to narrow it down this far already!)
