= Solution
For two distinct commuting <real scalar fields> $\chi_1,\chi_2$ in place of the labelled <fermion> pair, with interaction $g\phi\chi_1\chi_2$, there is no anticommutation sign on closing the loop and no <spinor> <trace>. The two <propagators> are simply multiplied. In terms of physical time-ordered scalar contractions $C_1,C_2$, the connected contribution is
$$
\boxed{-g^2\int d^4x\,d^4y\,
D(x_a-x)D(y_a-y)C_1(x-y)C_2(y-x).}
$$
The <statistics and symmetry factors of a two-propagator loop> determine the replacement. Relative to the fermionic formula in <2/a/ii>, the loop-statistics sign reverses. Scalar source kernels normalized like the stated Gaussian give the same overall minus sign after the four contraction factors are converted. There is no spin multiplicity.
If both replacement fields instead mean the same real field $\chi$, the coupling normalization and the identical-line <symmetry factor> must be specified. With interaction $(g/2)\phi\chi^2$, the cubic vertex is $ig$ and its two-identical-line loop carries $1/2$, giving coefficient $-g^2/2$. With the literal coefficient $g\phi\chi^2$, its vertex is $2ig$ and the coefficient is $-2g^2$. These cases are distinct from two separately labelled real fields; suppressing their combinatorics would hide a factor-of-two ambiguity in the replacement.
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