In manual mode you still have all verification methods. Have a look at the specification at draft-ietf-acme-acme-01 for more details.
The LE server has to access your server like any normal visitor. And he has to use the normal domain (example.com) for this. The reason is that this is one easy way to verify that you are really the owner of the domain - that's the aim of every validation challenge. That's why you cannot push the LE server a different IP/domain to connect to, because the thing it wants to verify is not that you own a (random) server,/domain but that you own exactly the server/domain for which you want to get a certificate.
As I already said another method is to use DNS validation where this challenge does not happen at the server level, but before that.
This method and the webroot domain verification method are AFAIK the only ones, which you can currently use without restarting the server.
Apart from that you can just run the LE client on the real server (and IMHO it would also be the easiest way) instead of doing this on some other server.
Maybe it would be a good idea for you to tell us why you actually want to use the Raspi and not the real server and we can look for more specific solutions than commenting on general things like this. Especially as this discussion at the end is not really much related to the topic implied by the headline and mentioned in the first post.