Excuse me as a noob, its just I read TI's Arm-M4F TM4C129x MCU series -> Datasheet , there is a interesting built in peripheral called EPI (External Peripheral Interface), don't know if its suitable for your needs.
Attached below brief summary of what EPI capable of, and highlighted few items that I "guess" maybe suitable, cmiiw.
Just FYI, the EPI has some major errata associated with it. Be sure to check before you even think about using it.
They have a reference design for a SRAM board that fits to the Tiva C Connected Launchpad. Shouldn't cost much to have made and it's a quick way to test if the EPI will work for you.
Unfortunately it didn't exist when I was working on an EPI based design. After spending weeks laying out a board, I stumbled upon the errata, which wasn't linked to from the product page, but a post on TI's forum. So I posted, asking if it was valid. They told me it may or may not be an actual bug, it could have just been a software problem but they couldn't be sure, but an engineer would contact me with a definitive answer. They never did.
So, I switched to a PSoC 5LP, which also has support for external memory, going so far to support custom non-memory devices, in hardware even! Though, their documentation leaves a bit to be desired, but I can forgive that as it appears to be a new feature.
Anyway, the EPI on the Tiva C did appear to basically work, and it's well documented in the TRM. You can even execute code from it! Sadly for me, it seems their engineers never anticipated that someone would want to use fast, high density SRAM as a framebuffer for the onboard Graphic LCD engine hardware... I mean, that's totally an unpredictable corner case, right? *Sighs*