After looking at the impedances seen in all the different configurations, I decided that the most recent result just didn't make any sense. So I re-took the measurement, and the impedance magically came out to 46 ohms (see below).
What I think happened was that I changed the frequency range of the sweep after I had completed the calibration of the impedance test kit, so the analyzer was extrapolating out to where I was measuring. Another thing I noticed was that the results were different depending on whether I used single shot or continuous triggering. This final result was done with proper calibration and using single shot the way I was shown to do it the first time.
Finally this business can be put to rest (unless there is reason to believe that 46 ohms is not close enough to 50 ohms---I'm not sure what sort of reflected power we are likely to be able to handle.)
RA: 46 Ohms is OK, but not too hot. How about trying out some trimmable components too? When you finish, I recommend filling the box with some kind of goo (like RTV) so that it doesn't all rattle around. That's what the professionals seem to do.