To calculate the minimum required slew rate for the opamp, you have to start with a spec. You need to decide what exactly you want out for a certain specific input. This is not clear from your verbal description.
Since slew rate is something that happens over time, this spec must necessarily have a time component. One way to do this is to show the input signals and the desired output signal on a plot of voltage as a function of time. Ideally, the input is a step so that the rest is only about delay of the circuit itself.
After that, you can probably see how far the output needs to move within what time. The slew rate is the ratio of those. Note that this is one thing that is amplitude dependent. If you want a larger output, then you need a faster slew rate to transition between the two levels over the same time. Therefore, your example inputs and outputs should be for the largest output swing you want.
You mentioned time constant, but I'd look at this dV/dt across a capacitor, which causes a fixed current. During slew-limited transition, you probably don't have a exponential decay anyway.