Hi there,
I know you asked for a European freelancer, I just wanted to let you know I'm excellent at administering servers and I know exactly what to do with prestashop.
If your server has SSH access, that could speed up the process by more than half, since I could make use of zip or tar tools to compress all your files before moving them to the new server and then decompresing them. Also to restore your database via mysqldump. If you do not have SSH access, I can move your website anyways. It will be an slow process but I asure you it will be moved exactly as it is to your new server, without loosing any data and your shop will look exactly the same with all your plugins and configuration in place.
Send me a message, I have all day to work on your project if you choose to work with me.