Let $\lambda = t^{3/2}$. Then $t^3 = (t+1)^2$, and $\lambda^{300} = t^{450}$. Examining the equation $P(x) = x^3 - (x+1)^2$, it has a unique zero at $x=t$, and since $P(2) < 0$, we have $t > 2$. Let $a, b$ be the other two roots of $P(x) = 0$. Note that $|ab| = 1$, and $a$ is the complex conjugate of $b$. Hence
$$|a| = |b| = |t|^{-1/2} < 2^{-1/2}.$$
Now take $M = a^{450} + b^{450} + t^{450}$. Since $a, b, t$ are the three roots of the monic integer-coefficient cubic equation $P(x) = 0$ and $M > 0$, $M$ is a positive integer. Moreover:
$$|M - \lambda^{300}| = |M - t^{450}| = |a^{450} + b^{450}| < 2 \cdot 2^{-450/2} < \frac{1}{4^{100}}.$$
This completes the proof.