"(5) ... SQLite allows multiple processes to have the database file open at once, and for multiple processes to read the database at once. When any process wants to write, it must lock the entire database file for the duration of its update. But that normally only takes a few milliseconds. ..." - https://www.sqlite.org/faq.html

You can start your new version of your application in a new process that opens the same database file, switch your load balancer to the new app, allow the old to drain all requests and then terminate the old app.