Monitoring a large array of load cells
It will probably work just as you suggest.
As to how to determine the individual load cell numbers, you have four
switch positions, so you'll generate four equations.   There are four unknowns.   Measuring, quicker than the situation changes, with all four switch settings, you can solve. The four values measured will be related to the 'pure' input values in a linear way, so it's a matrix equation. The matrix inverse acting on the measured values, gives the load cell values. One wants to know the noise and signal sensitivity (this will come from an eigenvalue analysis).
Do the math FIRST, if the matrix isn't invertible, that's a show-stopper. 
You'll need to worry about electrical pickup of the wiring, since your amplifier
gain will be applied at some distance from the load cells.