The agreed value part of a policy should be easy. Insurance is based on risk. It is for you to tell the insurer what the risk is. Clearly some of that is contentious and insurers protect themselves with complex legal terms of business. If they catch you lying, you will loose value for the deception. The value of the thing is not huge in the range of risk, unless it is undervalued or overvalued. If undervalued, the insurer has no problem. If overvalued the insurer might choose to challenge despite accepting the contract. Of course the value is a component of the value of the deal so a higher value means a higher premium.
So a proper agreed value policy will ask for the valuation. It used to be that insurers asked for several pictures, an independent valuation or the invoice for the purchase of the vehicle, if very recent. There is no reason not to to supply those item anyway and clearly mention them in the correspondence. It is then beholden on the insurer to reject that evidence. An invitation to send the premium means they must have accepted the documents. Remember that is docs first, payment second, otherwise your back up information is not binding. Normally an insurer will come back if they are unhappy with a valuation. That is clearly more likely on a vehicle they have never heard of.
Always pays to check out the small print. What is the status of the car in transit not being driven? At a show where it is unattended on public or private ground. That sort of thing. Can be some nasty surprises. Fav is folk bombing about on a field and driving into each other to find they are not insured. I had that with some long hairy git leaping in my KR175 without so much as a by or leave and driving into Bob Cotton's Bond MkA, I think it was, driving along minding its own business. Who pays?