How did Pauli and Fermi deduce the existence of the neutrino? From Wikipedia:

The neutrino was postulated first by Wolfgang Pauli in 1930 to explain how beta decay could conserve energy, momentum, and angular momentum (spin). In contrast to Niels Bohr, who proposed a statistical version of the conservation laws to explain the event, Pauli hypothesized an undetected particle that he called a "neutron" in keeping with convention employed for naming both the proton and the electron, which in 1930 were known to be respective products for alpha and beta decay.[6][nb 2][nb 3]
n0 → p+ + e− + νe James Chadwick discovered a much more massive nuclear particle in 1932 and also named it a neutron, leaving two kinds of particles with the same name. Enrico Fermi, who developed the theory of beta decay, coined the term neutrino (the Italian equivalent of "little neutral one") in 1933 as a way to resolve the confusion.[7][nb 4] Fermi's paper, written in 1934, unified Pauli's neutrino with Paul Dirac's positron and Werner Heisenberg's neutron-proton model and gave a solid theoretical basis for future experimental work.

Can you explain why beta decay could not be explained by adding that tiny amount of energy (attributed to the neutrino) to the KE of the emitted electron?
 A: The momentum is harder to deal with than the energy.  If a stationary neutron decays into an electron with momentum $(a,b)$ and a proton with momentum $(a,-b)$, then there is no way to conserve 3-momentum without the creation of a third particle with momentum $(-2a, 0)$
And while real numbers wouldn't work out this nicely, it would be obvious that the proton and the electron had one of the components of their velocity moving in the same direction in a cloud chamber.
A: An electron is a charged particle, charge conservation would not work as the neutron has zero charge. In addition it would have been detected with its interaction as its energy would be similar to the energy of the other electron seen.
The neutrino was posited as a weakly interacting particle exactly because it was not caught by the detectors, and because energy and momentum conservation would not otherwise work for each event.
Edit after edit of question

Can you explain why beta decay could not be explained by adding that tiny amount of energy (attributed to the neutrino) to the KE of the emitted electron?

It is all about momentum and energy conservation. The neutron mass was known, the proton mass was known and the momentum measured and the electron mass was known and the momentum measured. It is easy to go to the center of mass system , i.e. where the neutron is at rest for the presumed two body decay. In the center of mass system the proton and the electron should have equal and opposite momenta which constraint defines also their energy in the center of mass system, one unique value. Instead the data showed that it was not a  two body decay but a three body decay, since there was a distribution for the energy and momenta of the proton and the electron. A zero mass spin one half particle balancing momentum and energy solved the problem. 
A: It's all about spin.  The conservation you mentioned is the key to it - you could conserve energy and/or momentum by tweaking the resulting KE, but you're still left with unconserved spin.  Historically, observations showed an upper limit to the electrons which was not high enough to include the missing energy that is found if you don't include neutrinos.
