Convergence of off-shell loop integral

Given the following theory

$$mathcal{L} = frac{1}{2}dotvarphi^2-frac{1}{2M^2}(Deltavarphi)^2-frac{m^2}{2}varphi^2-frac{g}{4!}varphi^4$$

it’s easy to show that the free propagator ($g=0$) in Fourier space is given by

$$G(p^0,p) = frac{i}{(p^0)^2-dfrac{|p|^2}{M^2}-m^2+iepsilon}.$$

I want to compute the one-loop corrections to the coupling constant $g$ in the off-shell configuration, so setting all external momenta to zero. Being this a sort of $g varphi^4$ theory (which i imagine being regularized by the presence of the $M$ term). We know that there are three one loop corrections that are related all to the same loop integral. In the off-shell configuration this integral is of the form

$$intfrac{d^4k}{(2pi)^4}frac{1}{left((k^0)^2-dfrac{|k|^2}{M^2}-m^2+iepsilonright)^2} tag{1}$$

In my mind this integral should converge to something along the lines of $m^2/M^2$ or $m^2/M^2$, but it’s just a gut feeling. By computing it with the help of wolframalpha I find this feeling to be correct. Then I try to solve it in the following way

$$int dk^0d^3kfrac{1}{left((k^0)^2-dfrac{|k|^2}{M^2}-m^2+iepsilonright)^2} = M^3int dk^0d^3k frac{1}{left((k^0)^2-|k^2|-m^2+iepsilonright)^2} = int d^4k frac{1}{left(k^2-m^2+iepsilonright)^2}$$

where in the second to last step I used the change of variable $kto k/M$. But I know that this integral does not converge since, if we regularize it, we get someghing along the lines of

$$int^Lambda d^4k frac{1}{left(k^2-m^2+iepsilonright)^2} propto logfrac{Lambda^2}{m^2}$$

What’s wrong in my evaluation? How should the integral $(1)$ be computed?