I hadn't realised quite how close to me you were 😂
I do think it's price, but also timing. You can probably hold out for the price, but the market here dies until March or so - and to be honest, you'd probably want to take it off the market this month, so it didn't look like it'd sat unwanted for months before then when you relisted.
Character houses do sell for more around here, and the racecourse is a great area, it's super close to town. Chase Meadow is a proper "estate" and a walk to town, but families love it, for the pub/parks/proximity to Warwick Parkway. My first flat was there, a one-bed for £1k per month.
If the cute cottage is the Guys Cliffe one, it's the A46 and the constant roadworks that's affected the price of that...
But I don't think it's necessarily that your price is awful. You just need someone to love the house, and not many people are looking right now, especially with all the local roadworks. It probably isn't helping that Rightmove's "nearby houses that have sold" are all on your street, but have sold for £210k, £307k, £298k, £295k, £220k; and then within 0.1 of a mile, £210k, £260k, £200k... Even if people can see that yours is worth more, it is going to make them think that yours is at the top end of what your street can sell for, which means there isn't much possibility for house values to go up... and as you've said, there's also not much potential to extend yours, so there's no way to "add value", and Leamington and Warwick are mad about that in the same way that Stratford is mad about making everything look like it was designed by Shakespeare himself.
I'd take it down for a few months, if it was me. Our EA told us to list our home in Warwick over Christmas in 2020, as "Rightmove traffic goes right up". It does, but it's nosy people at home with families, not serious buyers. We took it down, relisted on the 1st April and sold 10 days later. Three doors down kept theirs on, but ended up getting £60k less than us because buyers wanted a bargain for a house that "wouldn't sell", and they got no full price offers.