= Solution
In <phi cubed theory>, a two-point <bubble diagram> has two cubic vertices, two internal <scalar propagators> and <symmetry factor> $1/2$. Its connected amputated insertion is $+\mu^\epsilon g^2 B_d(p)/2$. The inverse <scalar propagator>, or quadratic <one-particle-irreducible vertex>, therefore receives the negative insertion:
$$
\Gamma_E^{(2)}(p)=p^2+m^2+\delta Z\,p^2+\delta m^2-\frac{\mu^\epsilon g^2}{2}\int_0^1dx\,I_{6-\epsilon}\big(m^2+x(1-x)p^2\big)+O(g^4).
$$
Equivalently, the connected two-point correction is $G_0(p)^2\mu^\epsilon g^2 B_d(p)/2$, and expansion of its inverse changes the sign. The same sign follows from the second term of $\tfrac12\operatorname{Tr}\log(K+\mu^{\epsilon/2}g\phi)$.
Using the six-dimensional residue and $\int_0^1x(1-x)dx=1/6$, the divergent inverse-propagator correction is
$$
\Gamma_{E,\mathrm{div}}^{(2)}(p)=\frac{g^2}{(4\pi)^3\epsilon}\left(m^2+\frac{p^2}{6}\right).
$$
Its <momentum> term requires <wave-function renormalization>, unlike the massive quartic tadpole. The <minimal-subtraction two-point counterterms in cubic scalar theory> are
$$
\boxed{\delta Z=-\frac{g^2}{6(4\pi)^3\epsilon},\qquad\delta m^2=-\frac{g^2m^2}{(4\pi)^3\epsilon},\qquad\mathcal L_{E,\mathrm{ct}}^{(2)}=\frac12\delta Z(\partial\phi)^2+\frac12\delta m^2\phi^2.}
$$
These are additive coefficients in a Lagrangian expressed in terms of the renormalized field. If $\phi_0=Z^{1/2}\phi$, $Z=1+\delta Z$, and the bare mass is separately written $m_0^2=m^2+\Delta m^2$, then $\delta m^2=m^2\delta Z+\Delta m^2$ at this order, so $\Delta m^2=-5g^2m^2/[6(4\pi)^3\epsilon]$. Distinguishing these conventions prevents a spurious disagreement between mass <counterterms>. The local Minkowski <counterterms> are $\delta Z(\partial\phi)^2/2-\delta m^2\phi^2/2$. A complete cubic theory also has one-point and three-point subtractions; they are not the specified two-point bubble poles.
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