Many s the cat that has a permanent diet of Synulox because the leucocyte count stick was positive. Specific gravity is also not reliably tested by the dipstick method. You will just have to buy a refractometer or send it in to the lab. Sediments are best examined with the urine still warm. Remember that crystals will precipitate with cooling so sediment examined in the laboratory will always show a higher crystal count than the urine as it emerges from the urethra.
Crystals also precipitate as pH changes.
Proteus infection or contamination of urine can precipitate struvite crystals in a spectacular manner. If it is an infection, treat that first before prescribing diets. Bacterial culture can be done on a free flow sample if you accept that the results may reflect the contamination. Mixed infections are also more likely to be contaminants. However if there is dysuria present, the natural defences of the urinary tract are de facto impaired, so it can be useful to know what is around and what it is likely to be responsive to.