I'd suggest moving to a database that supports network connections and just use the DBI package. I'm sure MySQL and PostgreSQL would be good. If your company already supports other databases, you might use one of those. I assume you're already using DBI to access SQLite, so if that's the case you'd just have to build your database on the database server and figure out the connection strings for DBI.
On the other hand, if your application doesn't require much in the way of data, and you don't expect it to grow beyond the capabilities of SQLite, then I guess you could always use DBD::Proxy so you can run SQLite on the database machine and your Catalyst app on the web server machine.
When your only tool is a hammer, all problems look like your thumb.