I'm not sure how to use SQLite via web browser. If you mean Web SQL Database, it seems unsupported as well today. IMH, IndexedDB is the best choice. Theoretically its storage up to 50% of free disk space so you don't need to worry about the capacity.
However working directly with IndexedDB is quite complicated and time consuming. So I like the wrappers at higher level such as localForage or JsStore, Dexie. Use localForage if you like simple key-value set/get syntax. Go with JsStore, Dexie if you need more complex queries.
I'm also interesing with a new kind of database: the ones that sync automatically. A few names such as GUN or PouchDB. They are a combination of storage and socket IO. They may be suitable for applications that require a lot of real-time interaction.