Briefly the question here is whether there is or could be a theorem analogous to that of Littlewood for semiprimes (generalized prime numbers which are products of two primes, repetitions allowed), and so a Skewes-type number for semiprimes. In other words, whether we might have
$$li_2(x)-\pi_2(x)=\Omega_{\pm}f(x)$$
for a function of x not too different from that in Littlewood's theorem.
The graph below is of the absolute difference between $li_k(x)$ and $\pi_k(x)$ for $k=$ 1-14, 1-16, 1-18, 1-20 for $x=2^{14},2^{16},2^{18},2^{20},$ the [almost invisible] green, ochre, violet and blue lines, respectively, in which
$$li_k(x):=\int_2^x\frac{ (\log\log t)^{k-1}}{\log t\cdot(k-1)!}dt. $$
I hope it is clear enough where $li_k$ comes from. Just as $li(x)$ is a better estimate of $\pi(x)$ than $x/\log x,$ $li_2(x)$ is a better approximation of $\pi_2(x)$ than the corresponding prime counting function for values I have checked. More to the point, there is no Skewes theorem for $x/\log x$ which for $x\geq 17$ does not exceed $\pi(x)$ (see Wiki article on the prime counting function).
The x-axis begins at k=1 on the left, and (for example) the first value for the blue line is $li(2^{20})-\pi(2^{20})\approx 110.5,$ so while it is difficult to see, the first values are all $>0$, as we expect, since $li(x)>\pi(x)$ until the first Skewes number (about $1.397x10^{316}$, according to the Wiki page on this topic).
Parenthetically, I think we can show that the lines (joining points at k=2,3 in this picture) cross the x-axis (negative to positive) near the mean number of factors of $x,$ which is asymptotically $\log\log x.$ Also, the sum of the points below zero is algebraically equal to the sum of those above zero.
So for an x just beyond Skewes number the graph changes sign (if I have calculated correctly) near 3.8 and the first point at k=1 is now negative, and it is tempting to speculate (and well beyond the scope of the question but included for context) that the entire set of points changes sign.
