First the reference. All the precision reference IC's allow you to attach a potentiometer for calibration. Add this with a link so you can go from the factory calibration to your calibration. There is every chance your meter calibration is very good. I would test the reference and if it looks like it reads within the manufacturers specs, then adjust it so it reads 10.00000V on your meter. The thing is most of the time, an error of 0.01% is not a big deal. What you want is all the ranges of the meter to match, so if you had 10 of these reference chips all adjusted to be exactly 10V according to your meter, then if you put them in series, you want your meter to read 100V exactly.
Now in picking the chip, look more at temperature stability and long term drift rather then accuracy. I would prefer a 1ppm/C 0.1% reference chip to a 5ppm/C 0.02% reference. I cannot go looking at data sheets right now - doing some hard disk recovery work, but Mouser and Digikey searches are a good starting place.
Secondly, you need to get a feel for how much resistors vary when you have a meter like yours. Grab a metal film resistor - perhaps 10K, but the value does not matter. Measure it on a handheld DMM. The reading will probably look absolutely rock steady. Now put it on your dmm. Watch the values drift. Just the sub-milliwatt heating of the resistance test is enough to make the resistor change with a 6 1/2 digit meter. Touch the resistor with the tip of a finger, and you will really see the value change. So This is why it is so important that every resistor in the 100V divider is dissipating the same power exactly. Now when you close the switches in the Harmon divider, this is no longer the case, so you have to change your technique. You let it stabilize first with the switch off. Close the switch and adjust the output to be exactly double the reference (as described before) and open the switch. Let it settle again. Quickly switch the switch on and adjust again (you have to do it in a few seconds). Keep repeating until the initial output voltage just after you turn the switch on is exactly double the reference. Then you will know that with the switch off, you have an accurate 100V.
Third, the opamp. By slow, I meant that a 1Mhz (or even 100K) opamp will probably be easier to get stable then a 10MHz opamp. Here is the story about stability. In a feedback circuit, there is exactly a 180 degrees phase shift between the output and the opamp input connecting to the feedback at DC (ie you are connecting to an inverting input). In my circuit, we connect to the + input as the transistor is inverting the output. As the frequency increases, capacitances causes gain fall offs and at the same time, phase shifts. If the sum of all the phase shifts reaches 180 degrees and the gain is equal or great then 1 from the opamp input to the circuit output, the circuit will oscillate.
Basically if you have the output connected back to an inverting input, and the circuit has a 180 degrees phase shifts due to capacitances, then the signal the input sees is now in-phase (non-inverting) compared to the output and instead of negative feedback, you have positive feedback. Positive feedback is always unstable if the gain of the feedback loop is 1 or greater. Positive feed back means that if the output drops a little (you added a load) the signal the opamp input will see will say "drop the output even more", and this is the start of an oscillation.
So the trick they use in internally stabilized opamp is they add a single RC (resistor-capacitor pair) fall off that swamps out all other capacitive effects. So above a few Hz, the opamp has a 90 degree phase shift from this internal RC for AC all the way down to the opamp's unity gain point and so it is stable with just about any load other then a straight capacitive load (which could add another 90 degrees shift and make it unstable). The opamp designers do not care of the phase shifts below unity gain get really ugly as that does not affect stability. Now to boost the output to 100V, I have added an extra gain of 22 times, and the transistor circuit will have some phase shift of its own. This means that we can get oscillations even down to the point the opamp has a gain of 1/22, and as I mentioned before, the opamp can look really ugly for phase shifts down there. So you will probably have to add some extra frequency compensation to get the circuit stable. If you are into LTSpice simulations, it is a good exercise, but otherwise, playing with the 100P capacitor in my circuit is a good starting point.
Hope that makes some sense.
Richard.