How to run two Messenger applications logged into different accounts? I have done a bit of research, and that, coupled with some programming knowledge, leads me to believe that it should be perfectly possible to run two separate applications of Facebook Messenger.

Is it as simple as changing the manifest file name? Or is there more to it? How do I run two separate Messengers?
Yes, at the very least that will have to be done. You may also have to re-sign the apk, and also change any absolute references to the Application context inside the java code and/or machine code. This will require lots of effort, you might as well build your own Messenger interface via Facebook's API. Facebook does have a developer API for this kind of stuff, you may want to look into that.

Facebook Developer SDK

I would try renaming the package first, without even decompiling anything, if it crashes due to internal package pointers, I would probably try to build a plug-in.

Hope that helps,