Damage insurance in a hotel (UK)

I am currently staying in a hotel in the UK, for business purposes (I have been invited to a university for research,and they booked the room for me).

During my stay, I have caused by mistake the bathtub to overflow, which resulted in some damage; namely, the hotel had to relocate me to another room, remove the carpet and let it dry, which caused them to have to pay maintenance and be unable to rent the original room for 5 days. This is what they told me, and sent me an invoice, asking for me to pay as soon as possible.

Now, as I understand it (from my experience in France and what I have been told when asking people from the US and France), either my, the university's, or the hotel's insurance should cover this. I have been told by my insurance not to pay myself for the damage, but let the hotel (or its insurance company) deal with them directly.

However, the hotel manager insists that I pay myself before leaving, and then handle that with my insurance (without the hotel interacting with my insurance at all). He then stated that this was how things were done in the UK, and that they had nothing to do wth my insurance company or the university's.

I am obviously quite worried about this, as it sounds like there is a non-zero chance doing so will end by me paying for the damage, and never managing to get reimbursement by either insurance company for lack of documentation of the incident.

Is it really how things are supposed to be done in the UK? If not, what should I do?