Ahem... If it's a total loss, then you really shouldn't have a splitter for sale...
I'd ask them to clarify what they mean by "evidence of cars that I feel reflect the true value of mine". They might still be after evidence to value your car - so service history, MOT certificate, etc... Offer them photocopies or email them scanned copies for the purpose of valuation - they don't need the originals for that. Obviously, when you sign the car over to them, you should surrender the originals, butdon't part with them until you have to.
From my previous experience with Adrian Flux, my dealings with them were very brief. Like "here's the insurers phone number, now go and hassle them yourself" kinda brief.
As someone else said - don't cash the cheque as that could be seen as accepting their offer. Again from the incident mentioned above, the claims assessor went and took a look at the car, assessed it, decided it wasn't worth fixing and gave me a call with the offer. It was, frankly, insulting and I obviously turned it down. Get ads from anywhere to support a higher valuation - I went looking for comparable cars (model, age, mileage) on autotrader for example. Having battled the price up, the decision was no longer quite so clear-cut anymore, so the assessor offered me the money or my car back repaired. I took the repaired car. On reflection, if I'd known how quickly the bodywork on that side of the car would deteriorate once the 3 year warranty on the repairs was done, I might have taken the cash...