Another question still floating around - do we really need the composite integrator? Is an OPA140 (for example) not enough?
This is a good question. I think it depends, as even the OPA140 will leave a small approximately square signal at the input. Depending on the reference current and cap this can be some 10 mV or so.
The amplitude should be +- I_ref / (2*Pi*GBW*C). So with a 1 nF cap., 10 MHz GBWand 300 µA Iref this would be some +-5 mV
If the impedance stays really constant with reference switching, that is good resistor matching for the positive and negative side, one should get away with a single OP integrator. With some 0.1% resistor matching the 2.5 mV residual square wave would correspondent so to some 5 mV/14 V *0.1% or around 0.3 ppm of INL error from this effect. With a smaller capacitor the error could go up and down with a larger.
A 2 OP integrator will make the circuit more tolerant to resistor mismatch, as the residual voltage would be only fixed length pulses, that would ideally not cause an error even if the resistor do no match.
There is one more possible point: the OP for the integrator gets a input voltage dependent power dissipation (should be highest with an input near zero) and thus could cause a small thermal effect. with the 2 OP integrator the DC critical OP is the "slow" one and this one does not get mach variable power.
A downside of the 2 OP integrator is likely slower settling to a current step. This would mainly be an issue with really fast modulation that usually comes with a small integration cap. The circuit for the 2 OP integrator is that way that one can leave out the "slow" OP and get the 1 OP integrator as a fall back. So a layout could easily alow for both options

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So a 1 OP integrator (e.g. OPA140, OPA1641, maybe OPA827 - expensive but faster) is a real option if the resistor matching is good and the integration cap not very small.