Can a L293D H-bridge be used for this application?
We log the voltage drop over a resistor when the current passes through it.


The L293D is a transistor H-bridge. Using this to switch the current direction is not the right way. There are much smaller and better FET based integrated H-bridges available to do such polarity reversal.

However, the above method is a solution to a problem you do not need to solve. If your measurement supports both current directions, the problem is also resolved. You can use a shunt amplifier, such as INA199, to measure bidirectional shunt current.