Personally I'd have taken your line, its us that encourage them after all. I'd dare say management are more than aware of the issue but perhaps a letter in their direction? Having recently returned from India where, at my hotel I watched a bloody enormous rat run behind a piano at a very nice, clean restaurant I appreciate the issue but I just kind of shrugged... but hey, that's India.
A young rat ran past my foot in the Woolpack garden a few weeks ago. Not bothered. The important thing is that food is stored safely in the kitchen and waste food is stored securely before collection. That's just about all that can be done.
I had occur once in a pub near the river and agree there is no need to name which one. Even the cleanest kitchens/ restaurants/homes can get mice at times especially near the river or when road or building works are carried out. I choose not to mention it to the management at the time as I didn't want make a fuss as it was a friends birthday. However the next day I emailed the pub and told them. They emailed me back very promptly thanked me for letting them know , apologized and told me what they were doing for pest control.
So personally I would do that, again if i were you, if they do not get back to you with an apology and assure you they are dealing with the problem, then I would report them to environmental health department.