This kind of assumes some stuff that might not be true.
1. The database is some open source project like Postgres or Mysql and you can easily grab containers for it from dockerhub.
2. Somewhere in the code there is a schema file for the database (assuming it is a db that has schemas) so you could at least init a local copy of the database schema (although it wouldn't have any data in it, you would have all the tables) in your docker-compose setup.
3. Once you have the app and the db in docker-compose then come up with a way to seed the db with some dummy/test data (not a full dump of the real database, as you mentioned it is quite large, just enough data where you can see things working in the app).
You do that and show your manager, and they will quickly start throwing more stuff at you. You will also gain a reputation as some who takes initiative.
Good luck.