How to drive a sensitive, constant-current, 1MHz resistive device?
1 - The first thing you need to do is rethink your choice of amps. An OP37 simply will not respond in nanoseconds. Neither will an AD625. And that 100 pF feedback cap is great for stability, but it also reduces bandwidth. If you want low-voltage, high-speed op amps, it's time to move away from DIPs. And you do want high-speed amps. Your current source needs nsec response times if it is going to respond to nsec transients in load impedance.

2 - For better results, put the 100 ohm resistor to ground, and float the sensor. That way, you won't have to worry about common mode response to your sensor fluctuations.

3 - I take it as given that your control voltage is not actually +/- 15 volts.

4 - If your sensor is ESD sensitive, the last thing you want to do is to connect/disconnect the sensor while the circuit is hot. Just don't.

5 - If you wish to drive 1 mA through 650 ohms, you are going to get 0.65 volts - and that is incompatible with your 400 mV number. Even so, you should consider diodes that are not optimized for relatively large currents, such as the (1 A) diodes you're using.

EDIT - Furthermore, I'm curious about how you're measuring circuit performance. On the one hand, if you're getting a lot of noise, what bandwidth is it? Is it AC line noise? And just as importantly, what sort of scope do you have that will let you detect 1 nsec, 0.1 volt spikes, which you say can kill your detector? This is pretty sophisticated work.