So here's what I have imagined at the moment (hopefully someone can tell me if this is actually feasible).
I'm currently planning to use an SN754410 for my H-bridge ( allowing me to pass through the higher voltage in either direction) and three HEF4067B dual-voltage demultiplexers, which (I think) can handle the voltage I need and work with the current flowing in either direction. (If someone could confirm this for me, I'd really appreciate it.)
I've laid out the wiring here. Ignore the wire that isn't actually connected, I ran out of pins with the Uno and didn't want to rewire it to a Mega just yet. Additionally there probably need to be some resistors along the lines to the flip dots.
The red and black wires carry the normal 5V and the grey and orange lines carry the (flippable) 7.5V, which pass into the demuxes' common input/outputs (one line into the demux controlling the rows, the other into the two controlling the columns). The white and yellow lines control which direction current is flowing through the grey and orange lines, and the green lines control what the demuxes have selected.
The H-bridge has an enable pin (1,2EN) that, when low, places the inputs and outputs in a high-impedance off state. Similarly the demuxes have an ? pin that, when high, places all the outputs in a high-impedance off state.
Currently I have these wired in the diagram to permanently enabled, but I'm not sure if the Atmega will switch the 1A and 2A inputs synchronously enough that there isn't a brief time where the 7.5V input is shorted. Similarly I'm not sure if the ? pin can be safely unused and rely on synchronous pin changing to select the right output.