Portable client tool to connect to SQL Server Express LocalDB and execute a query
I suggest just writing a simple .NET console app to do just that. It can accept parameters for "ConnectionString" and "Query". This will be flexible enough to use in various situations.

using System;
using System.Data;
using System.Data.SqlClient;

namespace SimpleSqlCmd
{
    class Program
    {
        static void Main(string[] args)
        {
            using(SqlConnection _Connection = new SqlConnection(args[0]))
            {
                using(SqlCommand _Command = new SqlCommand(args[1], _Connection))
                {    
                   _Connection.Open();

                   _Command.ExecuteNonQuery();
                }
            }

            return;
        }
    }
}


Usage would be:

SimpleSqlCmd "Server=(localdb)\InstanceName;" "Do some SQLs;"


UPDATE:

I have created a project on GitHub to host a more functional / complete version of the code shown above. The code on GitHub will be updated over time whereas the code here will not be updated (since that is really not the purpose of this site). The repository URL for Simple SQL Exec (renamed, to avoid confusion, from "SimpleSqlCmd" as it is shown above):

https://github.com/SqlQuantumLeap/SimpleSqlExec