Story feedback - Doppelgangers and a race against time

So here's something to consider--
What if the PCs don't wear the ring? If they cannot identify the ring, they may well take it to an appraiser or sage in town who can determine its lore. If the sage mis-identifies the ring as some minor magical item, the PCs may just sell it off. If the sage identifies it as cursed, certainly nobody will wear it.
At least in the above, you are in a civilized area so your arc can go forward-- you may just have to get an underling or sibling or thief put on the ring and receive the curse unawares. Maybe that's an interesting tact- a talented pickpocket makes off with the ring & wears it, but now the doppelganger has rogue skills to perform assassinations. The PCs must now determine for themselves whether the cut-purse is only slightly or very guilty as the murderer.
Alternatively, perhaps a noble seizes the PCs store of magical goods when they enter town. Some munis do have an adventurer tax. That would make for an interesting story-line, where noble and ignoble versions of the same influential NPC are on the loose and PCs must intervene to save the city leader.
One more thread, what if an evil character puts on the ring? Won't the doppelganger be pure? This doesn't really affect the stakes but I don't really understand the motivation for the magical being other than mayhem, and if it wants to bring about the death of the cursed ring-bearer there are far easier ways to be dispatched than being arrested and executed.
My only point, there are a few spots in this arc where your ingenious characters can derail the plot entirely. Good to have a back up plan, or use it as a bottle adventure, a short arc you can spring into effect at any time by having the ring stolen by an NPC.

Now a thought, research the cursed ring of Rudolph Valentino. He was an actor who met an untimely demise after bearing a likely cursed ring that ruined his career, probably killed him, and perhaps left him a wandering ghost. People and locations connected to the actor are subject to bizarre hauntings and grisly deaths.
Famous Cursed Ring of Valentino
Perhaps the doppelgänger is a manifestation of the ghost in the form of the ring-bearer. This works particularly well if the bearer is a bard, sorcerer, or other charm-type. The murders could be echos or repeats of crimes and deaths exerted by past victims of the ring-- an important facet of any cursed item is its history, the woes befallen previous wearers should impart a dread gravity to the ring. If you want to loop the entire plot back on itself, either the alchemist or lord should be relevant to the history of the ring. That is, the ring should exert its influence to be found at an opportune time. If the lord has an order placed for some sort of wedding or festival or significant occurrence, that event might be what draws the cursed ring back into the known world. I imagine an heiress of the lord betrothed to an undesirable match, who wishes to know the passions of a legendary lover- it's her desire that calls the ring out of the wizards trove and sets into effect events that will lead to the misery of many. In this narrative, the alchemist is almost certainly related to the origin of the ring-- cursed rings don't just come into being, and having the alchemist a member of the sect that originally crafted the ring for some purpose could be a good back story than that this person just <happens> to have the cure for that very curse. Ghostly rage or malevolence would be a good rationale for the doppelgänger being so single-mindedly destructive.