When we were looking to buy a house, the house we saw first was 'Dream House'.... everything we saw subsequently wasn't as good. We put in asking price and got rejected and told that despite being chain free (we lived in rented), another couple had put in a higher offer which had been accepted.
We continued looking, offered on another house (which the Estate Agent kept saying 'Ooooooh, can you do just 5K more? Just 5k More? They REALLY want to move, but even though it's above asking... another 5K?) And after the 2nd time, we told the EA to forget it, we'd find something else (the house eventually went off the market unsold)
I was working in London and my partner phoned, said he had a phone call from the Estate Agent who had represented the 'Dream House' sellers, that the original buyers couldn't get a mortgage and did we want another viewing. I literally commuted 2 hours back home, went straight around to the house and after 10 minutes we offered again (higher than before).
Despite STILL not being the highest offer, we got it accepted on the condition we could exchange and complete very quickly. As we were chain free, this wasn't an issue......
Long story, short, we've been in our 'Dream House' for just over 2 years and we still absolutely love it and nothing else came close. Therefore I'd not give up hope, but wouldn't bank on getting the house.... write the polite letter, but keep looking - that way, you've done all you can and won't think 'What if?', but won't drive yourself crazy if it doesn't come to anything.
Hope it goes well 