I don’t think it will work. The display normally initialises on start up. Switching with the power running will likely lead to garbled display pixels. You’re also messing with the integrity of the datalines. DIN/DOUT might work, but you’d be sending the data right before switching = corruption.
A bit laborious, but you could send the data lines of J8/9 and the displays to additional buffer chips (541 or 125). (It would almost work with the buffers/595s already present, but there are extra signals to deal with.) The outputs of the buffers can be enabled by tying the /OE pins low. Meaning if they’re high, no signal passes. You then switch one datastream off and the other on, probably with a short manual delay in between. Resetting the Core would also not hurt, but could be okay without.