I think you can fix your original circuit by replacing U1B with an integrator.
Making U1A slower is counterproductive because it makes the feedback signal slower, so U1B is reacting to delayed information. Intuitively, if you are driving a car and the speedometer is delayed by 10 seconds, you would find it hard to maintain a fixed speed compared to if your speedometer updated in real time. Adding a capacitor to U1A basically does that and makes the feedback signal slower. If you're the driver of this car (like U1B), and you couldn't look out the window, then you would slam on the gas until the speedometer read 40mph (your target speed). By that time you're probably actually going at 100 mph. Then you would see that your speed keeps going up so you slam on your brakes, and the car reaches 0 mph. However, your speedometer is delayed so you would see it go to 100 mph and then down again. Once it reaches 40 mph and keeps going down you would slam on the gas again, and the result would be that you keep on oscillating between 0 mph and 100 mph instead of converging on 40 mph. If we got rid of the 10 second delay, you would probably be able to maintain 40 mph easily (though if you drove like U1B in this circuit you would slam on the brakes when the speed reaches 40.001 mph and slam on the gas when it reaches 39.999 mph, and your speed would still be unstable).
The scientific explanation is that op amps have a pole at very low frequencies in the single digit Hz range. The open loop transfer function of your system is the product of U1A and U1B's transfer function. Adding a capacitor to U1A adds 90 degrees phase shift after the pole frequency formed by the RC network. Let's assume the pole is at 1 kHz. Then at 10 kHz the phase would be pretty much 90 degrees, and the attenuation might only be 20 dB (I am just making up numbers here but the logic is the same). U1B in open loop would have a response similar to the image below (this is a different op amp but it's the same idea). You can see that U1B probably has a pole around 1-10 Hz, and therefore the phase is 90 degrees at 10 kHz. So at 10 kHz the phase is basically 180 degrees (and probably a little over because of random parasitics), which means it oscillates if the gain is high enough (180 degrees means the negative pin of the op amp sees the opposite movement of the output, so it will always try to move the system away from stability). U1B is basically open loop so it has a lot of gain (60dB from the image). The loss from the RC circuit is not enough to outweigh the 60 dB gain of U1B so it oscillates.
Making U1B an integrator means that it still has a 90 degrees phase shift, but at a certain frequency, the gain is under unity. As long as nothing else in your circuit has a pole around that frequency, your circuit is safe and won't oscillate. Maybe U1A has a pole at 1 MHz, but by the time the frequency goes to 1 MHz the gain of the integrator is basically zero, so there is no way your circuit can oscillate even if it can reach the 180 degrees phase shift.