The CS4100V-01L is not good enough for your needs in my opinion: -
Below 2 kHz the response of the CT will fall-off and, by 50 Hz you'll barely get more than a few hundred milli-volts from it with "several" amps flowing through the main conductor. See also note 3. It's telling you what a typical burden will be like and you are a long way from that scenario.
But, if you want the math here goes... The secondary inductance is quoted as being 20 mH. This allows us to estimate the primary inductance by dividing secondary inductance by the square of the turns ratio hence, primary inductance is 20 mH / 10,000 = 2 uH. Given that you have no burden resistor and, assuming your load is extremely light, means we can use the impedance of the primary at 50 Hz to convert current into primary voltage. In other words, there is no reflected impedance back to the primary.
It's operating as a voltage transformer now. So 2 uH at 50 Hz is an impedance of 0.628 milli ohms reactive. If the primary inductance had 10 amps flowing through it, the primary voltage would be 6.283 mV RMS. Given that it's a step up device, the output voltage would be 100 times higher at 0.628 volts RMS and not enough to overcome the forward volt drop of the bridge.
I also built this circuit with a 1:500 CST but only produced 1.4VDC
from the rectifier. Thoughts?
For a similar sort of CT tech, with 500:1 ratio, the secondary inductance might rise 25 times higher. This is because inductance is proportional to the square of turns. So now, the primary projected impedance will still be 2 uH but, you'll get more secondary voltage (5 times more). So, with 10 amps in the primary you'll get a secondary of 3.14 volts RMS. Less 1.4 volts for the bridge rectifier and you might see 3 volts DC.