The perturbation is not just ${\cal L}_{CT}$, but $-{\lambda\over 4!}\phi^4 + {\cal L}_{CT}$. The first term gives also infinite contributions and the counter-terms are added precisely with the purpose to make the difference finite. The first term $\propto \phi^4$ gives infinite corrections not because the physical $\lambda$ is infinite, but because such an "interaction" gives infinite corrections to the fundamental constants and without subtracting them the calculations results are useless.