AST and Rental Guarantee Query

Hi, I will be letting out my property via a Letting Agent to tenants who have been reference checked.

I wish to take out Endsleigh Rental Cover and they require certain criteria to be met, such as fully referencing the tenants. The Agency are saying that they have been fully referenced and it has come back Approved. They can't provide me with a copy of this due to Data Protection, but I can see it when I go in their office. In the (unfortunate) event that I need to use the Endsleigh Rental Cover, I am worried that I have no proof that they were referenced. The Agency aren't willing to give me anything in writing, and their Referencing Agency (Clearview) are only available via email. Alternatively, I can go via their own Rental Cover, which is much more expensive (over twice the cost), and apparently it is hassle-free.

What should I do about this?

Separately, there is a metal shed in the rear garden without doors. The letting agency is asking me to fix doors and secure it, but I insist it is a 'bonus' to the tenants to use IF they want to. Agency is telling me that I will be liable to pay any costs if they get injured if they trip into the metal shed.

Demands are being made at this very late stage, so wondering how I should tackle the Letting Agents. Any advice is much appreciated, thanks.