To calculating the overall gain of a Bainter stage, would I simply
work out the individual gains of the three op-amp sections. The
overall gain would then be the product of the three individual gains?
The short answer is: Yes, you can (probably) analyze them individually.
When asking what happens when you cascade multiple analog filter stages, the questions to ask are: what is the source impedance of the first stage, and what is the load impedance of the second stage? If a circuit stage has a large and complicated output impedance, then loading it with another stage can modify its behavior. When working with passive filters, this is a big problem: unless the load impedance of each stage is significantly greater than the source impedance of the prior stage, cascading passive filter sections will result in complicated changes to the behavior of each stage.
One of the attractions of op-amp based circuits is that op-amps generally have a very low output impedance; for the ideal op-amp, there is zero output impedance. Moreover, the op-amp inputs themselves generally have very high input impedance, ideally infinite. This means that circuit sections whose outputs are op-amp driven can generally be cascaded without having one stage change the behavior of another.
Consider this schematic of a Bainter notch (taken from an Analog Devices publication):
The "notch out" is driven by the output of an op-amp. Thus this circuit will have very small output impedance. In other words, the voltage at "notch out" will be relatively insensitive to the load that is connected. This output impedance will almost certainly be much lower than the input impedance.
Thus, in the design phase, you can analyze several cascaded notch circuits separately, and simply multiple their transfer functions together. After producing a design in such a way, you might want to simulate the entire circuit in SPICE to check for behaviors due to op-amp nonidealities, etc.