If at all possible I'd recommend upgrading your server; I'm inferring from the fact that you're unable to use "p4 integrate -3" that you're on a fairly old release. With the latest server release the "integrate -3" functionality is enabled by default, without having to use an extra flag or anything, per this blog post:
http://www.perforce.com/blog/130806/quality-life-improvements-renamed-files
Having this functionality is a pretty big deal since it makes the scenario you describe "just work" -- all you have to do is integrate and resolve as normal.
Without that functionality, you have to do some work -- either modify a branchspec to line the current versions of the files up and use that to integrate the content, or manually recreate the same rename operations in the branch so that the files line up normally.
Note that this functionality requires a new server in addition to a new client, so a 2014.1 client on its own (with an old server) won't do the trick -- a server upgrade is necessary. If you're on an old release you'll need to do a restore from checkpoint as part of the upgrade, per the instructions in the administrator's guide:
http://www.perforce.com/perforce/doc.current/manuals/p4sag/chapter.install.html#d0e1167
but I would recommend you don't let that deter you; you'll be getting a lot of improvements. :)