Access SQL Server Express LocalDB from local area network
LocalDB cannot be accessed remotely, it's by design and documented.

Here is a description of LocalDB.

Here is a list of LocalDB features and limitations.

I don't see how installing SQL Server can be a problem. 120 MB is not that big nowadays even for small applications. You can embed the setup in your application and ship SQL Express with your setup kit. If you need a SQL Server Edition with a small footprint, consider using SQL Server Compact Edition.