Assume that it is not possible to keep accurate track of the difference between how much fluid is placed in the bladder and how much is taken out, and that a "physical" measurement must be made (where "physical" can include using a sensor, or using some sort of actual measurement of the dimensions of the bag to compute the volume).
Here are some options:
Weight- Reliable if the density is known. Bladder can be hung from a load cell, or placed on a surface supported by load cells, and volume can be derived continuously.
Bag in tank - One easy way to calculate volume is to place an awkward shape in a tank of water and measure the increase in volume. Simply inflate the bladder submersed in water in a straight wall tank and an ultrasonic (or other) level transmitter will be able to accurately give you an continuous volume measurement. Archimedes' principle
Open Loop Calibration Curve - Reliable if the bladder is designed such that it always grows in a predictable fashion (not one side then the other). A proximity sensor or distance sensor (possibly ultrasonic or laser) would be used to target a location on the bladder that changes the most during fill and empty (center of a whoopie cushion for example). You would then take an empty measurement, record the distance, fill with 10ml, record the distance, fill with 10 more ml, record the distance, and so on... The cure you create can then be used in a computer or PLC to continuously derive your volume from the distance sensor input.
3D Scanning - I would only recommend this option if the others were not feasible, the geometry was unpredictable, and a continuous measurement was not needed. Multi-photograph reverse rendering, laser scanning, and LIDAR(as Inquisitive mentioned) are all possible options. The computer scans the object over a period of seconds or minutes then analytically computes the volume.
You may be familiar with LIDAR. This is how land topography is put into digital format to be used in various 3D models.
Is there some kind of "point cloud" device that could penetrate the bladder material but reflect off of the fluid contained within? If you could find such a device, you could create a point cloud of the fluid mass and calculate the fluid volume by importing the point cloud into volume/topography calculation software.
It would be like an x-ray machine that penetrates the bladder, but bounces off the water. The whole bladder would have to be scanned to create the point cloud of the fluid surface. You could then do a volume difference calculation between the 3D fluid surface and the bottom, flat, supporting surface.