Super interesting little speculative thread! This was something that was personally always weighing on my mind. Kind of a long-winded post, but wanted to share what I think is the most plausible reason for the Dark Blue Shirt Bart on the boxart. Making this a bit late as well tonight, so apologies if the formatting is a little messy.
To start with the blue shirt itself appearing in general merchandise, your post is actually the first I've heard of Bart's blue shirt being a leftover from Fox hastily sending outdated references to manufacturers. Neat!
I did a quick little internet search and it seems the source of this comes from a RebelTaxi video (which I had never actually seen!), but it doesn't seem to explain why the blue shirt was kept even after the show gained notable widespread traction. This appearance of Bart actually includes merchandise well beyond the early 2000s. This ScreenRant article here mentions the Bongo Simpsons Comics still using Blue Shirt Bart in some later issues. While I'm not super familiar with the Bongo Comics (I never read any of them outside of a Christmas-themed compilation one I had as a kid), there is a Bongo Comic compilation as recent as 2017 that prominently uses Blue Shirt Bart on the cover.
Abraham Simpson (Grampa) also suffered a similar issue in concept art/promotional art for the show. While he was initially depicted with a blue sweater in some of the earlier Tracy Ullman shorts, his pink-ish sweater was much more common in the series. I've seen two versions of the art used for Grampa's HUD icon in Hit & Run. One with the pink-ish sweater, and one with a more blue-ish color. The latter was the one Radical actually ended up using for Hit & Run, so we know this is official and not any sort of fan edit. The Blue Shirt curse also seems to have followed Grampa into the Bongo Comics as well.
I could see maybe Fox only really having these older Ullman models to go off of and that was sent out to those producing merchandise. However, the blue shirt was perhaps viewed as iconic in it's own strange way that it was intentionally kept and also made it's way over to SHAR's art? That's my only theory on why the Blue Shirt design was still popping up even in the later 2000s-2010s despite the show (and Bart's proper color palette) being very popular at that point. I feel the same might apply to Grampa as well.
You definitely wouldn't be wrong to speculate that Matt Groening likely wasn't the man behind Hit & Run's boxart too. We know from some very early footage from IGN and whatnot that Radical had plans of producing their own artwork for Hit & Run's mission briefings. However, a developer that popped by our forums 5 years ago (Noviwan) explained these were done away with because Fox had '(...) a specific team that does ALL 2D content and drawing (for authenticity reasons, which is totally understandable). That's why the game has those kind of odd graphical iconic mission-briefing screens.' Note the use of word 'team' there.
Because of this, I definitely wouldn't be surprised if the art in question wasn't handled by Matt himself, but rather a ghost artist that Fox paid to just mimic Matt's style. His signature was likely just thrown on the box for consistency/recognition? The artist might have not had too many references to go off of the game itself outside of the general theme, as not only is Bart's Ferrini noticeably off model, but the fire hydrant near the crashed car/Apu has the gushing water misplaced depending on what platform the boxart was on.
These errors also lead me to think of the (admittedly disappointing) conclusion that the Dark Blue Shirt Bart in Hit & Run's artwork was likely also an oversight as well. While I feel the blue shirt was intended from the start, a darker color was chosen by mistake. Given how Bart's lower body isn't super visible on the boxart, nobody caught it and it was just shipped as-is. That, or it wasn't deemed too big of an issue and left alone.
I'd love to hear more thoughts on the whole thing myself as I will admit this is the only time I've seen Bart depicted with such an off color palette for his shirt, so it's possible my theory of 'ghost artist just chose a bad color for Bart and nobody complained about it' isn't accurate.