Method for Sending Multiple (16) Channels of Duty Cycle Modulated Signal over Medium-Long Distance
Let's get terminology straight first. There are physical links between devices, i.e. communication lines. And there are logical communication channels. One physical link (e.g. RS-485 bus) can support multiple communication channels, however they will share link bandwidth.

What this means is that you cannot transmit different data to several receivers simultaneously. And your particular application does not scale up for time slicing since it requires pretty much continuous data stream for PWM to be effective.

So, if you want to control all your panels from one shield the only option is to find suitable physical link that can deliver 13 bit of parallel data at 800KHz over 25 feet. LVDS chips can be used for this, as @Umar suggested. Something like SN65LVDM1677, which gives you 16 differential lines. CAT5e/6 supports 4, so you need to run 4 cables to each panel. A bit cumbersome but easy to implement with just a few parts.

A second option is to serialize 13 data bits using something like DS92LV16
and send them over single twisted pair. This way you only need one CAT5e cable per LCD panel. In fact, you can mount two interconnected RJ45 jacks on each serializer PCB and daisy-chain up to 4 panels with one cable, each panel taking input from one twisted pair. IMO this is the simplest option.

Finally somewhat more complex approach is to actually create logical communication channels and instead of passing high-bandwidth PWM signal transmit only graphical data in RGB format. This requires you to have MCU like your BeagleBone at each panel location. The MCU will be connected to the panel with ribbon cable just like in your current development setup. These MCUs will be networked with "master" MCU telling them what to display.

With BeagleBone you have wide selection of networking options. I suggest using built-in CAN, but other bus interfaces can also be used with suitable extender chips (as in the articles from your links).