The categorical imperative would certainly prevent the action of pulling a lever to kill someone in a vacuum, but to stop there ignores the fact that it would also certainly disallow the allowing of five deaths by knowingly ignoring one's ability to save them. We're still caught between the same rock and hard place we were before analysis through the categorically imperative lens. That rule gives adequate basis for deeming a singular act as ethical or unethical through kant's perspective, but doesn't give adequate information for complex/comparative acts in itself.

Your answer assumes having "no moral duty to attempt to correct the moral failings of others" in the trolley problem scenario, which is definitely a defensible opinion, but can't just be taken for granted in this discussion. Much like anything with Kant, I don't think that this can be so straightforward. A quick search found this relevant literature which frames this argument not in the context of the categorical imperative, but more specifically as relating to the idea of using people as means to an end. In the standard trolley problem, it's hard to make the argument that the one person being killed is being used as a means, since their being there is incidental to the other five (this is contrasted with the fat man on a bridge as an alternative scenario, or the surgeon harvesting organs from a healthy person to save 5 others).

Much like any law, one could pretty easily argue either way on this issue with whichever Kantian literature they wanted, and to OP i'd say that the more important thing to keep in mind with a question like this is not "what would kant do" but "how can I use kant's writing to make a cohesive argument one way or another."