Consider the following simplified problem: I will pick two numbers from the non-zero integers with probability proportional to $ \frac{1}{|k|} $. So:
$$ p(|k|) \propto \frac{1}{|k|} $$
And we play the same game as above, I reveal one and ask you whether the other is smaller or not. Now, what is p(5)? This is just $ \frac{1}{R} \frac{1}{|5|} $, where $R$ is the renormalization constant. So:
$$ R = 2 \sum_{k=1}^{\infty} \frac{1}{k} $$
whoops!
So, now, what does this mean? is $p(5) = 0$? is $p(k) = 0$ for all k? Does this problem make any more sense if I took the probability proportional to $\frac{1}{|k|}^\alpha$ as $\alpha \to 0$? What does it mean to pick a number whose probability is 0?
How would I even write down these numbers to compare them? They're obviously much larger than the number of atoms in the universe...Do I have an algorithm to spit out the bits and try to compare them that way?
This problem is a syntax error. The premise that you can 'pick a random number' from all integers is invalid. You can't do this. The reason why your (faulty) logic appears to work is because you're implicitly capping the distribution from $[-M,M]$ and then waving your hands by taking $M \to \infty$, which can't be done.