I’ve seen this done twice, both successfully.
The first time was a friend. They missed out on the house at auction and 6 months later it was still the ‘house that got away’ in their minds. They approached the new owners with this explanation. The new owners saw the house as a stepping stone, they intended to live there for a few years, do as many improvements as they could, and then on-sell for profit. They came to an agreement where my friends paid the or their time, trouble and profit to date (they had done some work) plus an extra years worth or so.
The second house was mine. My house was up for rent, not sale, so I approached the owner via the Agent asking if they would sell. It originally was a no. I asked again a few months later - no again. A few months later they approached me to see if I was still interested and it went back and forth (with other potential buyers!) and took a long time and was incredibly stressful, but I was successful. I think I overpaid, but (like my friend above), I was buying for the next 20-30 years, and it was more important to me to secure the house, than to miss out by an amount of money I wouldn’t care about in 30 years time.
My advice? Be respectful, be quietly persistent and be prepared to pay a lot for your emotional connection. They may initially say no, but after a few weeks, the idea may marinate abit and they may change their minds.