Hadjamach, British Glass p 286-7 describes a electrodeposition method that would be appropriate. As it can take up to 6 hours just to coat it would seem an expensive method of decoration for a scent spray!
Duthie, 1908, offers other possibilities: Silver leaf - impossible here, Chemical deposition - possible, silver-solution paint - possible, interestingly he does not refere to electrodeposition for silver but he does for gold.
The chemical deposition is achieved with a solution of silver nitrate and this was used for mirrors, replacing mercury from about 1840, the process is carried out at a temperaure of 70-80 degrees Fahrenheit. It is followed by a backing process - which could solve the contact with fluid issue. Backing is first a coating of shellac and then a backing made from pigments, turpentine and size. It should be easy to determine if this was the process use, were it not for the small opening, as the backing should be obvious from the inside.
If you have a friendly surgeon with a laparoscope you will be able to see inside, or a chemist to analyse a small scraping from inside :shock:
I would expect that you could restore the missing coating by use of some modern silver solutions from art shops, try on a jam jar first. It would be superficial, relatively harmless and give an aesthetically pleasing result. It will also remain obvious on close inspection as the silver colour will not match the original exactly.
Replacing the coating is the other option but that would probably cost moer than the piece is worth.