Yes, Surpass will be able to help. How many databases you have will depend on how long it will take and how many they will do for free (because each database move takes time, they can't do 1000 5MB databases for each client that moves, but I believe they'll do the first 10 or so for free, regardless of size... and they have a rather acceptable cost for any additional ones).
What you should do is submit a ticket to the Surpass help desk, sales division (
https://desk.surpasshosting.com/) and ask them for specifics. This is a general forum with mostly user to user help, and while I can tell you that Surpass will be able to help move your databases over, I can't give you any details to how much help or how long it will take. You don't have to register for the help desk, but it will help keep your tickets organized if you do. And they'll be able to answer you in detail everything that needs to be done to get you moved over smoothly.
I hope you do move here, and you enjoy your stay at Surpass.