Buy a $5 mo instance in the Cloud VPS
Make a subdomain named dev.yourdomain.tld
point the 'A' in your current server's dns for that subdomain to the new server IP.
set the new server's dns to that subdomain and its 'A' IP
I would install Ubuntu 16.04 LTS others will argue their favorite os CentOS is another commonly used.
open a SSH session and change to root user
get to work
Digitalocean and Linode have tutorials search out what you need to do.
unix.stackexchange.com is a good source.
Google is your friend -- use it.
when dev.yourdomain.tld seems to be working to your satisfaction --
from that location -- I am assuming /path/yourdomain.tld/dev;
cd to [......]/dev
$ cp -R * ../
copy recursively the site files up a level to the main (parent) domain's directory
create a new server configuration for the parent domain on the new server
change your DNS 'A' for the parent domain on the old server
add the parent domain to the DNS new VPS server with the new server's 'A'.
Migration of the HTTP part and the MySQL is now complete?
Or, pay for a managed VPS server and and do the things that you are good at now.
Using a control panel on a VPS or dedicated server is like using a leaky condom IMHO -- You can't feel a thing and it leaks all over the place
