(England) Estate agent won't accept signed employment contract as income evidence, threatening to keep my holding deposit. Causing major distress. :(

I'm starting a new job next month, and as such I need to move across the country. I've had a holding deposit on a rental that seems great for my circumstances. I have passed all of the employment checks, and have been offered the formal job offer. They have provided me my contract, which I signed and returned to them. They say that due to technicalities involving the third party hiring firm they use to do onboarding/hiring processes, they legally cannot sign their side of the contract until I have started on my first day of work.

The estate agent have asked for evidence of income to support my application, to which I initially provided my letter of acceptance for the job, then when they stated that was insufficient as it could be forged, I provided a whole ream of evidence including printed receipts of the job offer, a reference from my hiring manager, my line manager, my signed employment contract, the online portal showcasing that I have been offered and accepted the position, and am set to start it soon, and email chains from my line/hiring managers explaining how they cannot provide a signed contract until I have started on the first day of work. The estate agent has now rejected all of this evidence and insists that I must provide a copy of my employment contract that includes both mine and my employers signature, despite me telling them repeatedly that I cannot do this do to the unique circumstances of the hiring process.

They now very curtly informed me that if I do not provide this "imminently", that they re-open the property to others, and retain my holding deposit. This would completely devastate me as the holding deposit was no small fee, and I had been reassured by the hiring manager, the estate agent, the referencing agency, and my line manager that they have provisions for this circumstance, and that I would be able to secure the accommodation without issue. I am now experiencing major distress and housing insecurity as a result of this. What can I do in this situation? Are the estate agents allowed to kick me off the holding deposit when I have already provided in my mind beyond sufficient evidence of employment?

Any advice would be appreciated, this has me extremely upset.