Even if she couldn't take or reduce a sight, she knew enough DR to keep or at least understand a track and knew that they had flown x many hours at y knots on a course of z degrees from Howland ("we must be on you"). She certainly knew they were on an island southeast of Howland and probably had a pretty good idea how far on what bearing.
For me, her "we must be on you" comment is evidence that she didn't understand what Fred was doing. Without an RDF bearing, the most Fred could know was that they had reached the LOP that ran through Howland. If they were bang on course, Howland should appear but otherwise there would be only empty ocean. I suspect that at some time after sunrise, while AE was trying to get Itasca to take a bearing on her, Fred passed her a note saying something like "ETA 1900" meaning that at 1900 Greenwich they would reach the LOP that ran through Howland. Amelia apparently took it to be an ETA for Howland. I don't think Fred would say "we must be on you." Without a three-star celestial fix there was no way for him to navigate to a specific point.
I'm not sure I fully agree with "Without an RDF bearing, the most Fred could know was that they had reached the LOP that ran through Howland." Here's why:
The crux of my argument is that determining latitude in the later stages of this flight was more important than longitude. If you're on the right course, you're gonna get where you're going (Howland) sooner or later and it wasn't like they were going to miss a connecting flight if they got in 40 minutes late. But if you're drifting right or left and you end up a couple of hundred miles (< 10% of total flight length) north or south, you might miss not only visual contact, but RDF range as well. Splash.
They were flying almost due East, hence their longitude was dependent primarily on their speed over ground and time aloft, while their latitude was dependent primarily on their ability to stay on the intended course, i.e. the left/right error of their flying.
First the longitude. With no drift measurement or other means of determining winds aloft, they could only determine SOG by measuring longitude and time aloft and then backing out a number for speed made good. This would then be extended along a DR track to compute future positions. So Lines of Position derived from landmarks or celestial observations (preferable LOPs that ran N-S) were crucial for determining longitude, SMG and ETA. Most important for ETA was the sunrise observation, as the sunrise LOP would be the last "easy" N-S LOP before the more difficult sun sightings were the only ones available. This is consistent with their flying the 158-338 line later in the day. It was the last good LOP they had.
Now the latitude. Any sighting of a northern star during the overnight would allow for a nice tight latitude determination. I don't know how visible Polaris would have been from their altitude (it disappeared at the surface between 1 and 2 degrees North of the equator) that night, but Capella, Hamal and Mirfak would all have been in the right place to provide a good lat fix. If a dawn fix was obtained on a star that would give them their latitude, then they only had a couple of hours of drift (N-S) uncertainty in their DR track by the time they got to the advanced LOP going through Howland.
I know they reported partial overcast at one point, but they only needed an occasional N-S fix to measure left/right error. With the built-in artificial horizons of the octant and the Brandis, all FN needed was to see the star itself, not the natural horizon.
Unless the weather had been a wipeout since the position they reported by radio during the night, I think FN had a pretty good idea where they were, not just that they had hit the advanced LOP. Before I'm accused of applying too much hindsight, I approached this from the perspective of "how would you try to find Howland if there were no RDF?". Whatever you think of FN as a professional, he had bet his life on this flight. You will have to decide for yourself to what degree he would have been willing to depend on outside assistance. My premise is that he was sitting there in a plane with the tools, techniques and experience necessary to do a lot more than just get AE close enough to Howland so that someone on the ground could bring them in. I think he used them to the fullest extent.
The USNO has an
online celestial navigation calculator for anyone who wants to see what was where when. (BTW Marty, Mercury was down, Venus was up and the moon was in a pretty good spot as well).
This is the data I was using for rough work: