I use Docker for running Calibre-web. Unfortunately, there is no management support for Docker in InterWorx. You could install it on the server, but InterWorx wouldn’t know about it or manage it in any way. Given the amount of work you’d be doing outside of InterWorx to get Docker up and running, it would almost be more worth it to just spin up a small unmanaged VPS on one of the many cloud providers that offers that sort of thing. Some of them can be started up for around $5/month.
Having said that, if you have root access to the server, you may be able to upgrade Sqlite yourself. I don’t know exactly the protocols that Liquid Web has in place, so you’d like need to clear it with them, there are articles out there about upgrading the sqlite version on CentOS 7. For instance: https://linuxhint.com/upgrade-to-latest-sqlite3-on-centos7/
I will say that this sort of upgrade should be done at your own risk. We don’t support packages/software provided by third parties (including CentOS), so this would be an unsupported change by us.
Even though this is unsupported by our support desk, if I get some time later in my shift, I will run through the instructions from that link on a test VM and see what happens.