How to export data from a SQL Server database with hundreds of tables and timestamp columns?
If the account that the SQL Service is running under has access to any network shares, then you can use database backup to a shared folder, and restore from there as well.

BACKUP DATABASE [dbname] TO DISK = '\\servername\sharedfoldername\dbname.bak'
RESTORE DATABASE [dbname] FROM DISK = '\\servername\sharedfoldername\dbname.bak'


Or you can use bcp.exe. Use SSMS Object Explorer to script out all the tables, and execute the script on the desired database
Use TSQL to build out all the bcp.exe commands, something like this:

select 'bcp.exe ' + quotename(object_schema_name(object_id)) + '.' + quotename(name) + ' out "\\server\someshare\' + name + '.bcp" -n -S ' + @@servername + ' -d "sourceDb" -T '  from sys.tables order by name;
select 'bcp.exe ' + quotename(object_schema_name(object_id)) + '.' + quotename(name) + ' in "\\server\someshare\' + name + '.bcp" -S "DestinationInstance" -d "destinationDb" -T '  from sys.tables order by name;