Problem with LT3750 charging
first of all make sure the voltage rating is D1 and C4 is much higher than 300V. For a capacitor, the voltage rating would be at least double of the operation voltage for safety.
Second, try to build a PCB to test your idea. I have experience to design a boost circuit using LT1930, from 3.7v to +-12v. I never be able to get a good output with breadboard, either very noisy or not enough output. After I deploy everything on a pcb board, the signal is so nice, clean and steady.

If you exactly follow the example in the datasheet, the best shot is using pcb. Please follow my suggestion: always using surface mount component(resistor,Capacitor), try to use solder paste for SMT. If you don't know how to use it, please search youtube "hotplate solder paste", you will find it is so easy. After I learned it, I very seldom use solder iron to solder chips. Believe me. It is amazing. The performance on pcb is so good. You will thank me. In your situation, you can start from easy one, choosing slightly big components for soldering.

By the way for starting of smd, you can check this video: https://www.youtube.com/watch?v=yHRabJupfSo
It is not a very good video, but good to start. I would find a chance to take the video if I populate a board next time.