This sounds like negligence, either the BS or the painters, they should be careful to warn customers. If it is proved then you can claim from one of them for specialist cleaning. If I were you I would go to the BS tomorrow with the jacket (don't try to clean it any more for now), explain the circumstances and ask for a third party liability claim form, at least make sure that they have a record of the incident. If it's not them then they will know who appointed the decorators and whether signs should have been up.
I used to work in a brewery insurance department and we often had claims from customers who'd had clothes damaged or stolen in the pubs - definitely worth a try, but don't do anything else to the jacket yet. Good idea to photograph both the damage to the jacket and the place where it happened, if you can.