I may be wrong here, and someone may know this product well. But I suspect you aren't going to get a sensible answer to your question about the quiescent operating current. Even if I could decipher the regulator part number, which isn't clear from the printing visible in the photographs, without knowing the value of all the other components it would be impossible to know the operating point and derive the supply current.
The only way I think you'll know is to buy one (or a few) and test it (them), and even then there's no guarantee that the quiescent current won't change with the next batch since it's not a published parameter.
Personally, I would look for alternatives to the sensor you're using to see if there's a ratiometric alternative (which provides an output proportional to supply voltage), or something with a faster start-up time. I see you're designing a flow meter, but what's the sensor?