How to vary amplitude of my custom pulse generator?
Use the TTL logic level signal you are currently generating with your 74LS123 to switch a 2:1 analog multiplexer between its two inputs. For each input of the multiplexer supply a precision voltage of the desired level. Make these precision voltage levels using op-amps hooked up as unity gain voltage followers using rail-to-rail op-amps. Create the two input voltages to the unity-gain buffers using a potentiometer wired between ground and your supply voltage, or better yet a precision voltage source. You might also include an output unit-gain buffer on the output of the 2:1 multiplexer if you find the output pulses are not well shaped when driving the load.

200 nano-secs is the equivalent of 5MHz, so you will need a pretty fast 2:1 Mux and output op-amp. Not unachievable for sure, but you must be diligent in finding the right parts for this application. The frequency range of the two input op-amps are not so critical. So you might get away with your LM358s for these two op-amps. 

Also, be certain to use bypass capacitors for supply voltages to all of the ICs. The 200 nSec pulses can be problematic in that they will challenge the ability of your voltage distribution wiring to supply the high frequency needed by the components to produce and drive this relatively narrow pulse with the 2K output load.

By the way the '123 series one-shots had their problems. You may want to try a '221 series - 74LS221, 74HCT221, etc.