Oh, this poor baby! Boy do I have "mucousy/mushy/bloody stool" experience, for FOUR months with my Marcel! He had giardia twice - which messed up everything; giardia, btw, is notorious for hiding under the microscope - so sometimes they can have giardia, but never test positive for it. If they do have it, meds are needed, reinfection can be very probable - and giardia can be very damaging to a puppy's intestines, it can actually start to digest the tissue.....and then...you've got months of mystery mushy/bloody/mucousy stools. This if often diagnosed as ibs/ibd/colitis and/or food allergies - and it is often a combo of the entire clinical scenario. So, has she been tested for giardia more than once?
If all tests are negative, then a food allergy is very likely. I'd get her off of Science Diet (and never go back) asap. My next question is - do you want to stick to kibble or are you willing to home cook? If you want kibble, get her on something like Orijen 6-fish, Natures Variety, Innova EVO, Canidae, Evangers -or- one of the "allergy kibbles" like Wellness, Natural Balance etcetera; I'm not the biggest fan of allergy kibbles bc they are always lower in protein, so they aren't the best long term solution.
IF you want to homecook (which would truly be the best solution to let her system rest), I'd recommend what Marcel's nutritionist did for him: a whitefish diet w/ pureed green beans and mashed white potatoes. This is a hypoallergenic diet - the "cleanest" protein source (meaning, least allergenic). Marcel had FOUR months of bad stools - his stools firmed up after 1.5 days, yes DAYS, on his new diet. It was a miracle, I tell you! And this was after a gazillion tests, a gazillion vet trips, and lots of vet bills, and agony for Marcel and myself! This isn't a long term diet - but it should give her system a chance to rest, AND to heal - bc as it's resting, you can give supplements intended to heal any damage done to the intestines.
I probably gave too much info here - hope I didn't overwhelm - please feel free to PM me as I understand exactly what you're going through. I cannot recommend Marcel's nutritonist highly enough - so please let me know if you'd like her info (she does phone consults). He has now progressed to raw, and she confirmed how great this is for him. BEST of luck and I hope for a solution for your baby!