First off I haven't been a senior partner in any installs so take any suggestions with a grain of salt, but it's an interesting problem. Does the new set of servers need to be accessible to users even during setup? If not, could you set aside a router that you make independent from the rest of the network? In this way you could connect the new servers to this new independent network and use the same IPs and hostnames etc. during setup.
You'd then be free to upload the databases and setup the new system. When it's up and running, for testing you could connect users' machines to this "dummy" network. When you're ready to conduct trials on the live network you could arrange a cutover phase where you bring down the old servers and then connect the independent network to the rest of your network. A big issue here would be data synchronisation between the old servers' database and the new, while setup work/testing continues on the new servers and the old servers continue to be used live. However since the SAP system, underlying database software and actual database will all be identical between the two systems at the point where you upload the database to the new servers, I don't see off the top of my head why you shouldn't be able to keep these synchronised by applying the redo logs from the old servers to the new on a regular basis. In this way when you're ready to go live on the new servers, you can arrange said cutover phase where you ensure that the data on the new servers is bang up-to-date (take down SAP and then do a logswitch on the old servers, and then copy over the last of the redo logs to the new servers). This whole concept is similar to what a lot of companies do to maintain their DR boxes as up-to-date as possible.
A big issue here would be what to do with new data if the worst happens and you're forced to fall back on the old set of servers. Again though, applying the redo logs from the new database to the old one should work, barring some serious issues with the database itself. But then, any such fundamental issues would have been discovered during testing.
I've never tried anything like this on a significant scale, and of course there could well be other complexities or stumbling blocks that I haven't thought of, but just thought I'd throw out the suggestion anyway to get the discussion started. Cheers,
SAP Basis Consultant