The informal use of “which”
Solution 1:
All three example sentences might have issues, but none of those issues are being caused by the use of which.
Consider the following rephrasing of each sentence in a way that makes it clearly grammatical, but while preserving the use of which:
- This isn’t a story. It’s a road trip, which is the same difference.
I've removed the second comma and replaced it with two missing words. It might be thought that the second comma was "standing in" for the missing words, but that would be an unusual interpretation. Whether the use of the same difference makes sense, is another issue. The use of which wasn't the problem.
- She’s going to take the test, which I really hope she passes.
I've removed the second comma and the final it. Again, it's not which that was causing the problem, but a comma. In this case it was also the addition of a word rather than the omission of two.
- (a) I have so much heavy lifting to do, which … I shouldn’t really be doing since my back hurts.
On the assumption that this is a single sentence, I have corrected it by removing that after really. The ellipsis simply indicates a pause in the middle of the sentence. Again, there is no problem with which.
However, if the original version was actually two sentences, then there would be no problem at all—aside from the ambiguity of the ellipsis and the capital I that makes it unclear if it's a single sentence or two sentences. If the pronoun is replaced with one that does not become capitalized when used mid-sentence, this becomes clearer:
(b) She has so much heavy lifting to do, which … She shouldn’t really be doing that since her back hurts.
Since She is capitalized, it's clearly the start of a second sentence. As a second sentence, the use of that is appropriate.
The only thing possibly wrong with this (stylistically) is if it's not actually dialogue but narrative. In dialogue, it's common to use ellipses to indicate the trailing off of thoughts (and speech) in this way. But it's fairly uncommon to use it similarly in narrative. However, even if used in narrative, it would indicate the intentional omission of words.