Computability of Kolmogorov Complexity of Turing-Incomplete language
The important thing to notice is that your programs always halt. This makes computing the Kolmogorov complexity an easy task, since given a string $w$, you can go over all possible programs of length $\le|w|+2$ to find the shortest program whose output is $w$. The upper bound $|w|+2$ follows from the fact that for any string $w$, &quot;w&quot; is a program whose output is $w$.
In fact, you could probably compute the complexity of a string in polynomial time, since compression is only possible in the case of repetitions (you can not take advantage of a more general structure). Something along the following lines should work $KC(w_1...w_n)=\min\limits_i \big(KC\left(w_{i+1}...w_n|w_i\right) +KC(w_1...w_i)\big)$, where $KC\left(w|w'\right)$ is just $KC(w)$ if $w'$ is not a prefix of $w$, and $KC(w'')+O(\log j)$ otherwise where $w=w'^jw''$ and $j$ is the maximal number satisfying this. The point is that you only need to compute the complexity of substrings of $w$, of which there are $O(|w|^2)$.