Divergence of the tree level scattering amplitude in quantum field theory
What you found is a very simple example of an infra-red divergence, which plague all physical theories with massless particles.
This kind of divergences are already present in the classical case (see for example ref.1), and they usually signal that you are asking an unphysical question, not that the theory itself is unphysical. For example, if you have massless particles it becomes meaningless to ask about the total number of them in a certain physical configuration, while asking about the total energy is a well-posed question. This is reflected in the mathematics of the theory through divergences: the first question results in a divergent expression, while the second does not.
These divergences are inherited to the quantum mechanical case. For example, IR divergences are ubiquitous in QED, but it can be proven that these divergences always cancel out for "meaningful cross-sections", i.e., for predictions that are well-posed in the sense of the previous paragraph. This is sometimes known as the Bloch-Nordsieck cancellation, or in the more general case of the Standard model, as the Kinoshita-Lee-Nauenberg theorem. See for example refs.2,3,4.
In your particular case, as you are dealing with a scalar theory as opposed to a gauge boson, the analysis is slightly simpler (although one must in general abandon on-shell renormalisation schemes if we are to have massless particles). This is discussed in ref.5.
For a further discussion of infra-red divergences, see refs. 6-8.
References
Itzykson C., Zuber J.-B. Quantum field theory, section 4-1-2.
Peskin, Schroesder. An introduction To Quantum Field Theory, section 6.5.
Itzykson C., Zuber J.-B. Quantum field theory, section 8-3-1.
Schwartz M.D. Quantum Field Theory and the Standard Model, chapter 20.
Srednicki M. Quantum Field Theory, chapters 26 and 27.
Ticciati R. Quantum Field Theory for Mathematicians, section 19.9.
Pokorski S. Gauge Field Theories, sections 5.5 and 8.7.
Weinberg S. Quantum theory of fields, Vol.1. Foundations, chapter 13.