I have a website hosted on a shared hosting environment and due to their service being substandard am moving the website.  They have a portal which would normally allow me to go through to a tool called mylittlebackup which used to let me create backups and download them.  Unfortunately when I decided to move, this tool has ceased to work properly.  I have been calling the company about 2-3 times a day to follow up on the ticket request to fix the access for it with promises of yes its being worked on now, yes we have escalated it to the admins etc. but its been about 2 weeks now with no results. 

The website is still functional at the substandard level it always has been so the database still works.  I have the connection string that includes the server name, the database name, username and password.  Is there a way to connect to the database and essentially download it into the equivalent of what would be generated by the likes of SQL Management Studio or in this case mylittlebackup?  I'm thinking a remote connection tool of the sorts should be able to do it but I'm not sure what one.

I do have access to a full MS SQL Server if that provides a way to connect to a remote database on a different server, and any other potentially useful Microsoft product if that helps.