The fact you're coming back to the same place despite looking at others in the area suggests it is worth a premium.
Either they're happy to wait and get the price they want, or they're not: Nothing you can do will change that. There's no harm at all in making an offer to them outright whilst you're there, you may well find that there's a common ground you're happy to meet in the middle at which you wouldn't get to if you make offers via agents. We did with our house, and got the deal we wanted.
Also as it's a home not a BTL, you need to consider just how much you want to live there. BTLs or investments can be viewed with cold eyes, whereas homes sucker you in! I'd pay above market for somewhere I really wanted if I thought it was perfect and could afford it, I don't get the hang up about people having to drive a bargain everywhere.
THIS THIS THIS. You absolutely must check this out first before getting too advanced, as otherwise you may end up stuffed.
As for the increase, very possible. Looking at the market, we've 'made' approx £70K in two years on our place, which is about a 35% increase. We did get a bargain though, so the exception rather than the rule.