How to determine matching coefficients in a Effective Field Theory?

Assume that I have the Lagrangian
$$mathcal{L}_{UV}
=frac{1}{2}left[left(partial_{mu} phiright)^{2}-m_{L}^{2} phi^{2}+left(partial_{mu} Hright)^{2}-M^{2} H^{2}right]
-frac{lambda_{0}}{4 !} phi^{4}-frac{lambda_{2}}{4} phi^{2} H^{2},$$
where $phi$ is a light scalar field with mass $m_L$ and $H$ a heavy one with mass $M$. Let the Lagrangian of the effective field theory (EFT) be
$$mathcal{L}_{EFT} = frac{1}{2}left[left(partial_{mu} phiright)^{2}-m^{2} phi^{2}right]-C_{4} frac{phi^{4}}{4 !}-frac{C_{6}}{M^{2}} frac{phi^{6}}{6 !}.$$

Assume that I have calculated the $4$-point function up to $1$-loop order and regularized it correctly (renormalization scale $mu$). The results are:
$$
begin{align*}
mathcal{M}_{4}^{mathrm{EFT}} &=-C_{4}+frac{C_{4}^{2}}{32 pi^{2}}[f(s, m)+f(t, m)+f(u, m)]
&+frac{3 C_{4}^{2}}{32 pi^{2}}left(log left(frac{mu^{2}}{m^{2}}right)+2right)+frac{C_{6} m^{2}}{32 pi^{2} M^{2}}left(log left(frac{mu^{2}}{m^{2}}right)+1right)\
mathcal{M}_{4}^{mathrm{UV}} & approx-lambda_{0}+frac{3 lambda_{0}^{2}}{32 pi^{2}}left(log left(frac{mu^{2}}{m^{2}}right)+2right)+frac{3 lambda_{2}^{2}}{32 pi^{2}}left(log left(frac{mu^{2}}{M^{2}}right)right)+frac{m^{2} lambda_{2}^{2}}{48 pi^{2} M^{2}}
&+frac{lambda_{0}^{2}}{32 pi^{2}}[f(s, m)+f(t, m)+f(u, m)].
end{align*}
$$

The matching at tree-level resulted in:
$$m^2=m_L^2,qquad C_4 = lambda_0,qquad C_6=0.$$
I would now like to perform the matching at one-loop, i.e. we again demand $mathcal{M}_4^{EFT}= mathcal{M}_4^{UV}+O(M^{-2})$.

Problem

We have two unknowns, $C_4$ and $C_6$, that need to be expressed in terms of $lambda_0, lambda_2, m, M, etc.$. But $mathcal{M}_4^{EFT}= mathcal{M}_4^{UV}+O(M^{-2})$ gives us only one equation.. I don’t see how we can determine both coefficients with only the above information.

Notes

I’m reading Adam Falkowski’s lecture notes, see here. In section 2.3, p.~24, he performs the matching with only the above information…