Product: Credit card

On the evening of Saturday, XXXX XXXX, 2015 my XXXX year old dog XXXX got very sick. I took him to my vet where she stabilized him for transport to an emergency vet clinic. At that time my vet diagnosed him with XXXX and he was critical. Upon checking him into the emergency vet clinic, they confirmed the diagnosis and advised the cost would be between {$3400.00} and {$4200.00}. I agreed to this cost and decided to fill out a credit card app for Care Credit. Knowing my credit score at the time was above XXXX ( it was XXXX at the time as confirmed with XXXX at XXXX on XX/XX/2015 ) I agreed to the credit card application. When I completed the brief credit application via a computer terminal in the vet office, the girl at the counter talked to me about the credit limit. Based on the doctor 's estimate, she arbitrarily set the credit limit at {$4500.00}. My dog had a rough go of it and ended up staying at the emergency clinic from XXXX XXXX on Saturday, XXXX XXXX until his release around XXXX XXXX on Wednesday, XXXX XXXX, 2015. This was an additional one day stay above and beyond the original estimate. When I checked him out of the emergency clinic, my total cost was {$4400.00}. 
On XXXX XXXX I received a credit alert from XXXX where I maintain a credit monitoring membership. It revealed a massive drop in my credit score as well as an over the credit limit notice. A few days later, I received my first bill from Synchrony Bank whereby they show the charge from the clinic of {$4400.00} and the interest charge of {$25.00}. This total of {$4500.00} was enough to put me over my credit limit ( an ARBITRARY amount set by the clinic ) and start the tumble of my credit score that I have worked so hard to maintain. 
After receiving my first bill from Synchrony Bank on or around XXXX XXXX I called the bank and spoke with XXXX and explained my situation. I felt that the amount set by the clinic was arbitrary and once they assessed the interest charge, I was over the limit and this caused my credit score to drop by XXXX ( XXXX XXXX!!! ) points. She advised she would generate a dispute report on my behalf. I said I am not disputing the charge, the charge is valid, I am disputing the fact that the credit limit was an arbitrary amount set by the vet clinic that was guessing at the cost. She submitted the dispute. I received a letter dated XXXX XXXX advising that their report to the credit bureau was accurate they could not reverse this damage. On XXXX XXXX I called Synchrony Bank again and spoke with XXXX who submitted another dispute and agreed to reimburse the interest charge of {$25.00} which remains as a credit on this account. On XXXX XXXX I received another letter stating they investigated and are unable to modify or delete my request. 
Please know that I am not disputing the charge. My dog was about to die and I was going to prevent that if I could. I dispute that the credit limit amount was arbitrary and based on my credit score of XXXX, I could have qualified for a credit limit for twice the amount I was given. The inability for a vet clinic to correctly assess a credit line should not be the downfall of my excellent credit standing. 
Thank you so very much for any help you can provide.