It’s only 8k, which compared to the total cost is not insane. Just don’t offer more than 8k less than you can afford.
As for whether they’d find it out – you’d risk getting done, because when you pay in such a large quantity of cash, the bank want to know where it has come from (and taking out 20k would probably ring some alarm bells in their fraud reporting systems too).
You’d also be asking the vendor to commit fraud, which they’d would be bloody stupid to agree to. Not to mention that it’s pretty stupid for either you or him to hang around with 20k in cash, there’d be a lot of trust involved in that transaction.
Also, just because you’ve been rejected for a certain amount, it doesn’t mean that you certainly won’t get an offer accepted in the end – that was the case when buying our house, just took a bit of time for them to give in.
Joe