I am literally screaming at my screen "Use Postgresql!"
SQLite is the protégé of PostgreSQL. That link is to a talk given by D. Richard Hipp (the inventor, and still primary developer, of SQLite) to PGCon 2014.
Furthermore, if you look at the SQLite wiki, you will find the lines:
SQLite uses PostgreSQL as a reference platform. “What would PostgreSQL
do” is used to make sense of the SQL standard. One major
deviation is that, with the exception of primary keys, SQLite does not
enforce type checking; the type of a value is dynamic and not strictly
constrained by the schema (although the schema will trigger a
conversion when storing, if such a conversion is potentially
reversible). SQLite strives to follow Postel's Rule.
CHECK constraints whereas MySQL (astoundingly) does not! MySQL's support for Window/Analytic functions and Common Table Expressions (
CTEs - also known as the
WITH clause) is only in Beta - PostgreSQL has had them for years!
Overall, technically, PostgreSQL is a vastly superior database.
MySQL is as widespread as it is due largely to luck and good marketing - it ran natively on Windows (PostgreSQL does now also) back in the mid-nineties, and went GPL before the first internet boom.
Even if this question were not specifically about SQLite, I would unhesitatingly (99.9% of circumstances) recommend PostgreSQL over MySQL - PostgreSQL also has advanced full-text capabilities [1, 2] and would also be a much better fit for this sort of functionality - out of the box (thanks to @a_horse_with_no_name) - no need for secondary tools like Sphinx or Lucene.
If you are making (ab)use of the "well known quirk" of SQLite that it "does not enforce type checking" - you will have problems when replicating - although you will have the same problems with MySQL also.