Should Beta Minus decay put an upper limit on the mass of a neutrino?
This has been attempted, however the energy released in a neutron decay is a shade under a MeV and the neutrino masses are probably below $0.1$ eV. The energy of the neutron decay simply cannot be measured accurately enough to determine the neutrino mass.
The closest estimate I kmow of is reported in Neutrino mass limit from tritium beta decay by E. W. Otten, C. Weinheimer, but their estimate is $m(\nu_e)\lt 2$ eV, so it's a fair way from the expected mass.
Here is the KATRIN experiment:
The KArlsruhe TRItium Neutrino (KATRIN) experiment, which is presently being assembled at Tritium Laboratory Karlsruhe on the KIT Campus North site, will investigate the most important open issue in neutrino physics:
What is the absolute mass scale of neutrinos?
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Predecessor experiments at Mainz and Troitsk were able to appoint an upper limit to the electron anti-neutrino mass of 2.3 eV/c^2. KATRIN, using the same measurement technique, will either improve this limit by one order of magnitude down to 0.2 eV/c^2 (90% CL) or discover the actual mass, if it is larger than 0.35 eV/c^2. This requires an improvement by two orders of magnitude with respect to key experimental parameters.KATRIN measures the neutrino mass in a model-independant way via ultrahigh precision measurements of the kinematics of electrons from beta-decay. To detect the subtle effects of a massive neutrino on the kinematics of the beta electrons requires on one hand the provision of a strong gaseous windowless Tritium source with well-known properties and precision control. On the other hand it requires a high resolution spectrometer (MAC-E filter) with large diameter (10 m) to analyze precisely the electron energies from the source. All major components are under construction or already in place and in the commissioning phase.
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Like many ultra-precision experiments KATRIN pushes state of the art technologies to its borders. Its wide spectrum of involved physics and engineering, varying for example from molecular physics to nuclear physics, or from cryogenics over vacuum engineering to material science, makes the special touch and fascination of such an experiment.
from the wikipedia article
The experiment began running tests in October 2016, with measurements scheduled in 2017.
Your idea is correct. However, tyical energy releases in a beta decay are of the order of MeV, while neutrino masses are in the range of an eV or so. Current experiments are not sensitive enough to give a definite value of the neutrino mass, but produce upper limits. The best data so far, combining the Minz and Troitsk experiments (using Tritium beta decay), leads to an upper limit of $2~\text{eV}$ (see the particle data book, http://pdg.lbl.gov/2016/listings/rpp2016-list-neutrino-prop.pdf, page 4).
KATRIN (https://www.katrin.kit.edu/) is a new experiment, currently being assembled and supposed to start measuring this year, that aims to improve the sensitivity by a factor of ten, down to $0.2~\text{eV}$. (Note that from neutrino oscillations, one neutrino mass must be at least $0.04~\text{eV}$.)