The good news is that it sounds like you basically want a generic watch with a custom dial. This is going to be really easy to source!
Your best bet, if you don't particularly care about a specific movement, would be to go through Alibaba or Aliexpress. Plenty of watch manufacturers will essentially let you customize however you want and then print and assemble a product run for you. Realistically, the only thing you're talking about customizing are going to be dial compenents.
Things will get a *lot* more complicated (pun intended) if you want to focus on specific movements and/or specific complications, you will have to look for either a Swiss or Japanese manufacturer, and we can continue that discussion if that's the case! Just know that you're looking at a significantly more expensive run of watches this way. Good luck out there!
Looks like you are looking for work. You are a logo designer. But I don't understand one thing. Why you are looking for watch only? A logo designer can design any kind of logo for any kind of company with the idea of the product and some colors and the name.
I think you too can design more than just a logo for watch. My brother is a logo designer too and he's being designing for lots of company.
I suggest you to check out MakersValley, it is an online B2B platform that connect designers and boutique with italian artisan and manufacturers to create their private label with no minimum orders, factory prices and a dedicated account manager.
At MakersValley, you can also browse their BE INSPIRED Catalog of white label samples made directly by over 150+ Italian manufacturers & artisans. You can order any of these white label samples then customize it to fit your brand, plus you can add your own label on it!
If you are looking at private label solutions, you can also upload your project idea and have the manufacturers bid on it. Check it out: makersvalley.net and email them to email@example.com