Hi, I think I would consider going for PHP7 for performance issues.
However you must know that some code is deprecated...
What I think you can do is create a dynamic IP in front of your server, then I think you can copy your server and upgrade a new instance. It would mean double pay for a motth or less I guess, but then you could just ponit the IP at your new server when you have tested your code on PHP7...
Current server: IP X
New server: IP Y
Current domain: DOMAIN Z
Then dynamic ip -> IP X
When that is done, you point your domain:
Domain Z > dynamic ip
And then you can run two VPS , you then upgrade your copy VPS and upgrade etc..
When it's complete you just change the dynamic IP to point to your new server instance IP (IP Y).
I'm about 99% sure you can do it like this :-) Hope it was understandable....