= Solution
Let $p$ be the total external Euclidean <momentum> crossing one four-point channel. The two internal <scalar propagators> produce the <scalar bubble integral>
$$
B_d(p)=\int\frac{d^dk}{(2\pi)^d}\frac1{(k^2+m^2)((k+p)^2+m^2)}.
$$
Using the <Feynman parameter> identity $1/(ab)=\int_0^1 dx\,[xa+(1-x)b]^{-2}$ and shifting the loop <momentum> yields
$$
\boxed{B_d(p)=\int_0^1dx\,I_d\big(m^2+x(1-x)p^2\big).}
$$
In <phi-fourth theory> the <bubble diagram> has two quartic vertices and a <symmetry factor> $1/2$ from exchanging its two identical internal lines. There are three partitions of four labelled external legs into two pairs, the $s,t,u$ channels. The connected amputated loop diagram has positive coefficient $\lambda^2 B/2$, whereas the Euclidean <one-particle-irreducible vertex> in the effective <action> has the opposite sign. Thus
$$
\Gamma_E^{(4)}(p_1,p_2,p_3,p_4)=\mu^\epsilon(\lambda+\delta\lambda)-\frac{\mu^{2\epsilon}\lambda^2}{2}\big[B_d(p_1+p_2)+B_d(p_1+p_3)+B_d(p_1+p_4)\big]+O(\lambda^3).
$$
One can check this sign without external-line conventions by expanding the one-loop <quantum effective action> $\tfrac12\operatorname{Tr}\log(K+\mu^\epsilon\lambda\phi^2/2)$. Its second logarithmic term is $-\mu^{2\epsilon}\lambda^2\operatorname{Tr}(K^{-1}\phi^2K^{-1}\phi^2)/16$; four field derivatives give precisely the three negative bubble terms above.
The four-dimensional pole of $I_d$ is independent of its mass argument, so each channel contributes $2/[(4\pi)^2\epsilon]$. The divergence of the proper four-point vertex is $-3\lambda^2/[(4\pi)^2\epsilon]$. The <minimal subtraction scheme> therefore requires
$$
\boxed{\delta\lambda=\frac{3\lambda^2}{(4\pi)^2\epsilon},\qquad \mathcal L_{E,\mathrm{ct}}^{(4)}=\frac{\mu^\epsilon\delta\lambda}{4!}\phi^4.}
$$
For a Minkowski interaction $-\mu^\epsilon\lambda\phi^4/4!$, the corresponding local interaction <counterterm> is $-\mu^\epsilon\delta\lambda\phi^4/4!$. No momentum-dependent four-point <counterterm> is needed for these logarithmic poles. This calculation concerns the requested four-point bubble contributions; a massive two-point tadpole and vacuum diagrams require their own mass and vacuum-energy subtractions if those functions are also renormalized.
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