= Solution
In four spacetime dimensions $[\psi]=3/2$, $[\phi]=1$, and $[\partial_\mu]=1$. Hence
$$
[\bar\psi\gamma^\mu\psi\,\partial_\mu\phi]=5,
\qquad\boxed{[\lambda]=-1.}
$$
The coupling is an <irrelevant coupling> by <power counting in quantum field theory>, so it is a <nonrenormalizable interaction> that would ordinarily define only an effective theory with a cutoff. Here it is also a <redundant operator>: integration by parts gives $\lambda\phi\,\partial_\mu j^\mu$ up to a boundary term, and the <Dirac current> $j^\mu=\bar\psi\gamma^\mu\psi$ is conserved. This <derivative coupling to a conserved current> is removed exactly by the local phase redefinition $\psi=e^{-i\lambda\phi}\chi$.
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