This is a self-answer because I ended up analyzing Hickerson's data to get a couple of low-energy data points corrected for efficiency. My result is below, and is not endorsed by Hickerson and may be wrong.
Kappa is the energy of the electron relative to the maximum available in the decay. D is the probability distribution of kappa, normalized so that $\int D d\kappa=1$.
Only the lowest two points are corrected for efficiency (by me). The efficiency of the detector is nearly unity the higher energies. The line is theory. The horizontal error bars show energy resolution at some arbitrarily chosen points. The data points represent averages over this range.
The two low-energy data-points are ones that I arbitrarily picked at 65 and 145 keV. I corrected these for the efficiency of the detector from Hickerson's Monte Carlo simulation. Despite the extreme efficiency correction, the statistical error bars would still be too small to see on this scale, even down to energies of about 35 keV. At the 65 keV point, there is a systematic error in my analysis because the efficiency is rapidly varying, strongly curved, and concave up; therefore it's not a coincidence that this point lies a little above theory. As you go lower in energy, this systematic error becomes more pronounced, which is why I didn't include any points at lower energies. These systematic errors are a negative consequence of my method of analyzing the data, which I did in order to make the graph easier to interpret pedagogically. Hickerson presents his data as a comparison of experiment with Monte Carlo, which is better and eliminates the systematic error, but makes the graph harder to interpret.
After doing the things described above, I came across a pdf slideshow that includes the following graph, also from UCNA, on p. 21:
The graph is credited to B. Plaster. Although the statistical error bars are much bigger than for Hickerson's graph, this data set was apparently made with a much lower energy threshold, so the efficiency correction at low energies is not as extreme. However, there is nothing published about the efficiency, so it's impossible to correct it here. Unlike Hickerson's data, which has an absolute normalization available, this data set has no absolute normalization.