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I had some regret about paying what we did to secure our house at the time, and it nibbled at me for a few months after we moved in (mainly every time something broke or fell off).
But a year on it doesn't really matter – it's a desirable area, there have been precisely zero comparable houses come to market. What I imagine we overpaid is less than the £££ (and, oh god, the stress) of having to live in rental for a year/indefinite period of time waiting for another acceptable property.
Assuming there's some wiggle in your numbers to cover it, securing the house is step 1, trying pin the value so you can live with yourself is step 2 imo (and only an issue if your offer is accepted, hah!)
Just offer what you think the property is worth to you, within the current market.
Don't forget to appear credible by adding a sentence to explain your offer, and to appear prepared for the purchase by explaining your position. Obviously FTB is great, but even just to make it apparent that you're purchasing it to live in. Any context is favourable, some people like to know the buyer isn't an Air BNB landlord for example.