You didn't mention your DAM in text but since you speak about Darktable in title, I assume that this is your program of choice.

I think that (a variant of) 1. is the most recommended method since SQLite (Darktable's DB backend) allows (some sort of) multi-concurrency.

More technical details:

SQLite allows simultaneous reads but just one writer because the writer locks the entire file. However, since the writes in your scenario are quite rare and very very small (few miliseconds) and, more, you are just two users - you can use the same database simultaneously in a shared location.

Of course, you must be a little careful to not edit both of you the same picture. This is a constraint you must fulfill anyway since the edits are stored in XMP files and cannot be synced/matched automatically (if both of you modify the contrast to a file you with +10 and your wife with -5, who will win? - the last one?).

Another thing which perhaps will improve your workflow is to have separate storage(s) for the DB and images - this will improve the speed, depending of course of the size of your DB and files to process, especially if you have to work with big RAW files on a HDD (not SSD).

Yet another thing about syncing:

While syncing separate files - I mean here the XMP sidecars which hold the edits - can be a (semi-)doable task if we solve the trap above (who will win in the case of simultaneous/parallel edits on the same file), syncing the internal records of a DB can be a very tricky/impossible task, depending on DB schema and on what was added/updated/deleted.

just my2c & HTH