[III] - I ate the pie, and since then I have had a stomach ache.
I think this is the most natural comma placement: it divides the two independent clauses, which is what commas do.
I don't know who wrote your Sentence I (SI). To me, putting the two commas here is unnatural. It represents to me an unnatural way of expression. It also means there are two pauses within five words and two pauses in a straightforward sentence of 13 words. I don't know why anyone would want to slow down that much.
The two commas in SII make sense, because they separate a unitary phrase (since then) from the rest. But although this two-comma version works much better than SI, it's not as eloquently simple and natural as SIII. This is because SII also introduces an unneeded double pause in a thirteen-word sentence that needs, at most, one pause. (Note that my use of commas to separate at most is similar to II's comma use.)
You could also dispense with the 'since then' and write the remaining with either no comma or one comma:
[SIV] I ate the pie and I have had a stomach ache.
To me this expresses both thoughts as one unit containing two facts. It does not really stress a causal or resultative relationship between the two actions/facts. Also a comma is just plain unnecessary because now the sentence is unencumbered by the 'since then', and it has only eleven words. I mean it is short and consists of two short independent clauses.
[SV] I ate the pie, and I have a stomach ache.
This goes back to the simple connection of the two independent clauses with a comma immediately after the conjunction, as in SIII. This is very frequently done, and at least helps the reader parse the sentence if not also slow down a tiny bit. (See how I used only one comma in that last sentence, and only to separate the two clauses? Oops, I just did it again.) SV probably does not present the two actions in terms of one unit of dual-facts like SIV. Perhaps because the pause also allows a millisecond for the reader to make some causal or resultative assumption between the two clauses--even though the comma by itself does not do that.
Best Answer
The two words but and since are only accidentally together; they actually introduce different parts of the sentence.
But indicates the relationship of the main clause (I made another drawing) to what has gone before (It was a picture of a boa constrictor digesting an elephant). It marks a change of direction in the narrator's action, so it has to go at the front of the entire sentence as a sort of "hinge" in the story.
Since indicates the relationship of the subordinate clause (the grown-ups were not able to understand it) to the main clause: the subordinate clause explains why the narrator made another drawing.
The syntax would be clearer if you put the since clause in parentheses, like this:
But the author (or at least the translator) uses punctuation mainly to show the the narrator's speech rhythms, and this character doesn't speak like an academic delivering a complicated exposition with elaborate bracketings.