Dear Denis,
In a nutshell: the expectation of a sharp gamma energy holds only in the centre-of-mass frame.
If you boost to the lab, the energy of the photon is necessarily Doppler broadened (the intermediate ^7\textrm{Li}_\textrm{1st} is not at rest).
The kinematics indeed checks out. If you picture the process in two steps, in the first you’d have
\textrm{n}+^{10}\textrm{B}\rightarrow\alpha+^7\textrm{Li}_\textrm{1st}.
Assuming a 1 meV neutron and the ^{10}\textrm{B} at rest, the ^7\textrm{Li}_\textrm{1st} takes between 839.7222 and 839.7502 keV (triple check in case of arithmetic hiccups on my side in this quick check). Not a huge spread, but still that’s considerable kinetic energy in this scale. Proceeding analogously for the second step,
^7\textrm{Li}_\textrm{1st}\rightarrow\gamma+^7\textrm{Li}_\textrm{gs} ,
you see that the minimum possible energy of the photon is 469.999 keV and the maximum is 485.313 keV, which pretty much matches what you report.
So nothing mysterious, just good old Doppler broadening.
Cheers,
Cesc
PS: see also Prompt gamma spectrum from a proton beam - #4 by cesc