Ok, so before I took the eldest to school today I fired off the rpm install for bitcoin-qt. It works just fine with no modifications. Looking at the bug reports for OpenSSL for Fedora, it appears that they fixed all of the issues with bitcoin-qt but weren't/aren't interested in fixing what's necessary for bitmessage. So, in short... bitcoin-qt: works with the official Fedora OpenSSL libs bitmessage: works with the RingingLiberty OpenSSL libs, but requires a wrapper script both set the LD_LIBRARY_PATH variable correctly and ultimately actually USE the replacement libs.
Without manually setting the LD_LIBRARY_PATH variable, there is an "assumed" variable, for lack of a better word. The problem with setting this variable (for any reason outside of a wrapper script and via the way I posted) is that a) the behavior is inconsistent and b) it doesn't necessarily let applications that need libs outside of that path find those libs. Ultimately, if you were able to get an unhobbled OpenSSL 1.0.1x installed and working, you've got one on me because I simply couldn't, even with your instructions, get it to build. Using the instructions on the bitmessage.org website to install the replacement libraries in /opt/openssl-compat-bitcoin/lib and using a wrapper script that both didn't change the system variables (just for the app) and setting the variable correctly in the process (there's a website that explains it FAR better than I ever will here) allowed me to run both and run them simultaneously.
Thanks @DarkLight , I did a fresh install on my laptop this morning, and finished up tonight. Got BitMessage working on the first try. I think @DarkLight needs to make a new thread and I will lock this one, since it's no longer needed.