Firstly, I’m jealous because you have the best pub story ever: the scene in my mind is like mixing Crufts with the Sweeney. I wish you could sue Twinkle directly but I don’t know how she would give evidence in court. Would she put her paw on the bible? I digress. Yes you can claim as long as you can prove Twinkle’s owner was negligent. She has a duty to keep her dog under control and if she didn’t, she can be ordered by a court to compensate you for your losses.
As for paying them, it’s entirely plausible Twinkle may be insured and the insurer will pay. If not, her owner would be ordered to pay.