Car parking damage

Hello all,
An unusual advice needed.
I was going in staff carpark at the hospital where the boom gate was open and the car in front of me drove in without swiping. This has happened in the past where the boom gate has not been working properly and hence, swiping is not required. So, I followed the car in front of me but the boom gate closed on my car and caused damage. I prepay for parking so there was no benefit for me to not swipe. I purely didn’t to save time and the car in front of me went through. I assumed the boom gate was not working.

Now, the hospital is quoting me a policy saying they are not responsible for any damage to my car. They are essentially not responding to me regarding the footage showing the boom gate was open and was not working properly.

I just want to know what my options are here. I don’t want to claim on insurance if I have to pay the excess as the damage might get fixed for less than my excess and I don’t want to add to my insurance policy in the future. Would it be my fault because I didn’t swipe even if the boom gate was not working appropriately?

Thanks team!