My idea was to either use triggers or a process that reads Postgres's WAL and replay transactions to SQLite, by sending updates, rebuilding files entirely, or anything else.

Such replicator processes can be horizontally scaled, and made into an HA configuration.

Of course you should be careful to use on the Postgres side only the SQL features that will map well on SQLite. The range of such features is wide enough for practical applications though.

My real-life example would be a realty site, where changes are infrequent, but the desire to slice and dice the current large(ish) dataset in interesting ways is very high. The latter could be done using a per-node SQLite DB.