SQLite can have problems with databases exposed over the network depending upon how well the file-locking semantics are implemented by the file server/protocol in use. See https://www.sqlite.org/faq.html#q5 for more information.
Assuming it works, using a shared database directly like this can also lead to versioning problems. If you need to upgrade the schema of the database, you have to have some plan in-place for dealing with multiple versions of your add-on, etc. The simplest strategy would just be to switch to a new file/path whenever you issue a breaking schema change.
If you’d like to avoid compatibility and versioning issues, running a simple node.js app on a server on your network could work a lot better.