I've dealt with low volume manufacturing and prototyping...
If you are willing to do some legwork, you can contract it all out and get one for $2k… Maybe
Have someone design a pcb, another design the case, send the pcb plans to have made, send the case to a local machine shop or China to be machined or laser cut from whatever. For keycaps, you could hire an artisan to make you a set, it won’t be doubleshot thick pbt or ABS, but you would have them. You could also have someone design them in Cad and send them off to a company like Shapeways and have them 3d printed using their powder system ($$$).
That’s the good news.
The bad news is that it rarely works the first time, especially when coordinating so many people and companies, much less working with people and companies you don’t know or trust. Particularly China, where you run the risk of them just simply walking away with your money and you having no recourse. The keyboard creation tools Suicidal_Orange mentioned will help, but with you wanting custom sizing and such, it's probably going to need some manipulation in cad. Figure 6months to a year before you have what you want in your hands with 6 months being seriously rushed.
One offs and prototypes are not cheap, and shops tend to push them to the end of the line to make way for higher paying repetitive stuff so it could be 6 months before a shop even started on a case, depending on what you want. It's just how the industry works. I generally tell people $1000 to make just a case, and that is with you doing the leg work and if things go right, but it can go up to several thousand really fast, and you still would need your keycaps.
If you want production quality and be able to recoup your investment your 20k would get you either a few keyboards or the caps but not both. Honestly, the keyboard is the easy part, keycaps are tricky and require several molds. Molds EAT money like Pacman eats pellets. There is a reason good keycaps cost as much as they do, you need lots of rather precise molds, it’s time consuming and labor intensive.
The only chance you have of doing 100 or 1000 at this price is to coordinate with a Chinese manufacturer and leave them the rights to it to white label it to others (sell it to others), which you don’t seem opposed to. On the surface this sounds great, basically you subsidize their R&D. GREAT! Not so fast. The last guy I knew who did this sent a 3d printer over to have cloned, they quoted him a price, made his printers stuck them in a container and shipped them out on a boat. A week later, before he got his printers, mind you, those printers were on Ebay, Amazon and others for about $40 more than he paid, and they included shipping. Basically once he shipped them, he made nothing on each printer if he wanted to compete, 6 months later they were for sale for quite a bit less than he paid and again, with shipping.
For this, time depends on money and many other things, it could be 6 months or it could be 2 years.