Connect a Switchboard to Arduino
If I understand correctly:

You would want 4 on/off switches in parallel in front of each signal line from the strain guage. Each of the switches would be wired to one of the four quadrants of the wheatstone bridge. Only one (or none) of the 4 switches would ever be on. 

Repeat this for all strain gauges. 

Then you would just have to be careful what to only turn on 4 load cells (in general) for each quadrant of the load cell. 
 You would also have to calibrate every combination.

Something along the lines of this:
https://www.analog.com/media/en/technical-documentation/data-sheets/ADG1633_1634.pdf
or similar to do the switching. I have no experience with these chips, so maybe someone else can answer better.