LT3751 voltage never increases
You don't mention the most important things at all:

Values for R10 and R11.

For 500V output, R10 needs to be 715k and R11 needs to be 1.74k.

RBG is R9, and you have that properly a zero ohms.

All that is in the chart right there with the diagram on page 26:



The datasheet doesn't mention a minimum load.  But, since the chart doesn't go all the way to 0 for higher Vtrans it may have trouble regulating the output voltage for low loads.  Or, the fault pin is reacting to the load changes because you haven't set up the regulator properly.

Check and fix the configuration (R10, R11, and compare the rest of your circuit with the recommendations.)

Once it reaches the designed output voltage you can repeat your load tests.