You do not have to rely on chimes and your fickle friend, the summer wind. Weight driven clocks and bells definitely existed in medieval times.
The oldest surviving clock in England is that at Salisbury Cathedral,
which dates from 1386. A clock erected at Rouen, France, in 1389 is
still extant (see photograph), and one built for Wells Cathedral in
England is preserved in the Science Museum in London. The Salisbury
clock strikes the hours, and those of Rouen and Wells also have
mechanisms for chiming at the quarter hour. These clocks are large,
iron-framed structures driven by falling weights attached to a cord
wrapped around a drum and regulated by a mechanism known as a verge
(or crown wheel) escapement.
Your burly assistant lifts the weight a few times a day and its controlled descent plays a bell, or many bells. Or a music box-like tune with your theme song. Music boxes were invented in the 1800s but they could have made big ones with medieval tech: a turning cylinder with pegs that strike bells or pluck plates. Your assistant can sing along from time to time (he has a fine tenor) so everyone knows the words.
Another item which could be used is a whistling kettle. I cannot find any reference to whistling kettles before the 1800s which is weird; if someone can find one please link it in comments. You do not need high pressure steam to make a whistle; just regular steam. All you need is a kettle made the right way and a heat source. It occurs to me that in a low visibility world you would want a lot of lights, which means fire, and the heat product of fire is wasted when you use a torch or brazier or candle for light. So: put a kettle over it. Your less burly assistant can go out and fill it from time to time, with a following period of silence as it gets back up to boil.
Once you have pressurized gas coming out of your kettle, you could attach all kinds of different whistles. Water whistles sound like a warbling bird. I have one much like the below picture that I bought at the Renaissance Fair. From https://www.aliexpress.com/w/wholesale-bird-water-whistle.html
One more! This could be Dark Ages tech. Animal criers.
with some MS Paint.
I pictured caged parrots, crows and mynah birds acting as animal criers. But this cute little Bali mynah made me think: what if the animal criers were loose? They could drift around the area of their home shop, talking to passers by and/or singing their songs. I think this would be great for a story - talking animals are always a little surreal and eerie. Having the talking crier show up next to you and suggest something in quiet tones would be great. Having the same talking crier show up when you are far away from its shop would be even better.