LocalDB is absolutely supported in production. From the performance point of view it is identical to SQL Server Express, as they share the same database engine.
Now, for a site running in full IIS I would recommend using service-based instance of SQL Server Express. Since IIS is running as a service it is best to have a service-hosted database as well. That means you should not be using User Instance=true or AttachDbFileName=... in your connection string.
If you want to learn more, I recommend this post on using LocalDB from full IIS, part 1 and its follow up, part 2. They go into more details.