Interesting "bare metal" watch project. I think the hardest tasks will be sourcing a suitable display and managing the power budget.
Back in the mid 1970's, my father was involved in one of the early attempts at using LEDs to make a digital watch. (This was back when red was the only available color.) These suffered from low contrast in daylight viewing, and they required the user to press a button to momentarily turn on the display. Otherwise the LEDs would quickly drain the coin cell battery.
Digital watches became practical with the development of LCD displays. These are made with a thin layer of "liquid crystal" material sandwiched between two panes of glass, with clear conductive traces printed on the glass itself. Driving an LCD is fundamentally different from driving an LED: the drive polarity needs to be switched back and forth (AC not DC), otherwise the liquid crystal material gets an image permanently "burned in".
Getting a custom LCD glass is a bit costly because you have to pay the Non-Recurring Engineering costs (NRE), but for a watch application there should be suitable units already available in stock. Additionally many LCD display use triplex multiplexing. The logic to drive a triplex display is a bit complicated, but it's included in the custom timekeeping ASIC that the watch vendor would use. If you've never seen a digital watch teardown it's worth looking at; the ASIC is mounted on a PCB and "zebra strip" is used to contact the LCD glass directly. The whole thing is held together by the outer case. So if you're truly going this route, some level of mechanical engineering design is needed. It is usually also possible to purchase LCD glass that have metal contacts bonded to the glass, which can then be soldered to the PCB or plugged into socket strip.
You could try using an LCD Graphic Display Module, which has an integrated display controller included -- so all you have to do is send command codes. A quick search at mouser.com for LCD Graphic Display Module turns up this one from Newhaven display, needs 3V@1mA and also a 6V supply for contrast -- so you'd need a charge pump doubler, but could probably drive it from a 3V lithium coin cell for awhile... though read the CR2032 coin cell datasheet too. For a CR2032, 1mA drain is possible but kind of a lot. May be OK when active but display should be off during sleep. This kind of display has an integrated display controller that expects to receive command codes from a microcontroller, so it might not be right for your project.
Another technology worth looking into would be E-Ink Display (or E-Paper), which would seem ideal for daylight readability. Such displays are available through Adafruit and similar distributors, but they look a bit pricey. Also e-ink displays were originally developed for the Amazon Kindle e-book reader, so the displays tend to be more like the size of a tablet rather than the size of a watch. And, like the graphic LCD, you're more likely to find it as a whole integrated controller + display module, rather than just the bare display part. So again this might not be quite right for a true bare metal project.
Given the requirements of this project, I'd suggest this approach:
Research the internet for LCD display modules, LED displays, E-paper, etc. Most likely you won't find exactly what you need, but you can get an idea about cost budget, and which companies manufacture these units. Most important, see if you can find datasheets. (Must. Have. Datasheets.) Typical modern datasheets include web links to the manufacturer's website. Check the manufacturer's website, see if they have any way to contact them for applications support. (Note: I am an applications engineer at a semiconductor company, we love hearing what our potential customers are trying to do.)
It couldn't hurt to ask these manufacturers what it would take to make a custom glass, what the minimum lot size is, how much the setup and per-unit cost are... and ask if they may have something close to what you want, that you could use for a prototype. Only they can tell you what's possible for them and how much it costs.
You might also try contacting the folks at Sparkfun or Adafruit (who are more like hobbyists who became electronics distributors out of necessity), since this could be an attractive type of project (low-power, coin cell watch battery form-factor, simple/primitive/bare-metal display, maybe somehow "IoT" related...?). I think if you get the display design right, you could enable a lot of other similar projects.