I've found one of the problems in the code, and am now off by a (measly) two or so orders of magnitude.  In the meantime, I've repeated the analysis on the BBC events to see if any obvious problems appear.

The raw data:

Note that each step is clearly visible, where they could only be roughly seen for the BHT3 equivalent.

The offset calculation:

We see one strong peak, and a lot less noise than the BHT3, which we expect given the cleanliness of the base rate.

The fit:

The chisquared per degree of freedom is well under 1.  Note also that the background is extremely low.  The widths agree with the BHT3 values within a sigma or so.

After accounting for the prescale factor, this is .13 mb, which means we probably still have some missing factors.