Is my proof that a sequence of binomial variables of fixed mean don't converge in probability to the Poisson distribution correct?

You are right that convergence in probability is a stronger condition that does not hold here. Here's some intuition. If you simulate $1000$ draws from $B_n$ and $\Lambda$ and plot the histogram, you will see that they are quite similar. This is because $B_n$ converges to $\Lambda$ in distribution. In practice, this is a useful fact because Poisson distributions can be easier to work with.

On the other hand, convergence in probability requires that a realization of $B_n$ and $\Lambda$ are close to one another with high probability. There is considerable variability in both $B_n$ and $\Lambda$, and so there is no reason to expect every draw should yield identical values. This is what you saw when you calculated the PMF of $\Lambda - B_n$.

Here is another example that highlights the difference. Let $X_n$ and $X$ be i.i.d. standard normal distributions. Then $X_n$ converges in distribution to $X$ because they all have the same CDF. However, we can make $P(|X - X_n| > \varepsilon)$ arbitrarily large by choosing $\varepsilon$ small, so $X_n$ does not converge in probability to $X$.