Sql server backup is too big for express

Take a look at the actual file sizes & the Database Recovery model.
When a DB is in Full Recovery & the Transaction Log is not being backed up it will continually grow.
If you see you MDF file is (for instance) 1GB & your log file is 14GB then you need to backup & truncate the log file - refer the following MSDN article on Log File management
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Depending on what works best for you either of the following things will work.
1. Backup & truncate the Log file, then Backup your database
or
2. Set the Recovery model to Simple & then backup the database - this will work because once the transactions have been committed the Log file space can be reused when the DB is in Simple mode, if it is in Full mode you need to backup the Log file before the transaction log space can be reused.

Kind Regards

I'm picking up on

Quote:

it doesn't need to be an up to date copy just the structure,

Given that there are only about 3500 records have you considered just creating some scripts to create all of the tables then generating some script to populate the tables ... e.g. like Generate insert statements from a specific table data using SQL Server[^]
Microsoft do something similar to generate their sample databases e.g. Download Northwind and pubs Sample Databases for SQL Server 2000 from Official Microsoft Download Center[^]
See also How to: Generate a Script (SQL Server Management Studio)[^]
This is a method I have personally used and it worked ... as long as you don't need any recent updates and don't want to use GETDATE() anytime ;-p