Comparing the Kolmogorov complexity of theories
I'll try and give an answer to this question, and try to clear up some confusion as to the exact form of the question.

The first point I want to make: the $L$ in the statement of Chaitin's constant is indeed a function of $T$. In the absolute sense, it is monotonic in the expressiveness of $T$: if $L(T)$ is the smallest natural number for which 
$$T\not\vdash K(s)\geq L(T)$$
for any string $s$, then if $T'$ is a consistent theory stronger than $T$ ($T\vdash\varphi$ implies $T'\vdash\varphi$ for any arithmetic sentence $\varphi$) then $L(T')\geq L(T)$. The argument is very simple: if there exists $s$ such that $T\vdash K(s)\geq L$ then $T'\vdash K(s)\geq L$ by hypothesis.

However, this is only true if $L(T)$ is the absolute Chaitin constant. In particular, if $T'$ proves $\mathrm{Con}(T)$, then
$$ T'\vdash \exists L\forall s\ \lceil T\not\vdash K(\overline{s})\geq \overline{L}\rceil $$

by internalizing Chaitin's argument. However, a concrete $l$ for which

$$ T'\vdash\forall s\ \lceil T\not\vdash K(\overline{s})\geq\overline{l}\rceil$$

will in general not be equal to $L(T)$. In particular it may be much larger, generally proportional to the size of the proof of $\mathrm{Con}(T)$ in $T'$. This can easily be seen in the proof of the theorem itself, which crucially relies on the consistency of $T$.

So while $Q^*$ can prove consistency of the system with bounded induction, the length of these proofs gets longer the closer you get to $Q^*$ in expressiveness (one way to understand the incompleteness theorems is that the length becomes infinite when you reach $Q^*$, thus it has no finite proof of consistency in $Q^*$ itself). Thus the same applies to the various upper bounds on the internal $L(T)$s $Q^*$ can describe for each sub-theory.

So here is the short answer to your question: $L(T)$ is bounded uniformly for all subtheories of $Q^*$, but $Q^*$ itself cannot show that this bound holds for all such sub-theories. This was the crucial mistake Nelson made (buried under several layers of formalism) and that Tao pointed out here.