Hello, my name is***** and I have over 20 years of experience as a veterinarian. I am sorry to hear that Jasmine has had a rough time of trying to get bladder calculi resolved and that she continues to have blood in her urine.
It sounds like the most conservative treatment plan including an anti-inflammatory and prescription food, which often results in improvement and avoids the need for surgery, was tried and failed to fully address her trouble.
Ideally we like to take a series of radiographs as time goes on to make sure that the calculi are fully resolved with treatment. We cannot be sure of their resolution just by a lack of crystals in the urine unfortunately. Diets alone don't always fully dissolve calculi as sometimes the center of a calculus is composed of different material that cannot be resolved with diet. Those kitties need surgery.
Because they have fur, and don't bathe as routinely as we do, in animals a catheterized sample may not be sterile, or at least clean enough, that we can obtain a reliable, uncontaminated sample that is good for culture. The best way to collect a sample in animals is with a sterile needle and syringe to collect a sample directly from the bladder. This is not uncomfortable for the cat, is quick, and is much more reliable for culture.
There are specialty centers that use lithotripsy (sound waves introduced via a catheter while the patient is anesthetized) to break down calculi into small enough pieces to flush them out of the bladder. Unfortunately cats and small dogs are too small for this to be practical. We don't have small enough equipment, and breaking down pieces to the size they would need to be to pass through a cat's urethra could do a lot of inflammatory damage to the interior wall of her bladder. Standard of care for bladder calculi that aren't dissolved over this length of time is surgery to remove the calculi and to collect some to analyze them so we can prevent new ones from forming.
I understand it is upsetting to think of putting her through surgery, but long term getting those calculi will get her more comfortable, avoid long term inflammation and secondary infections and is best for her.
Please let me know if you have any further questions.