Blanks you are already in possession of? Or are you wanting them to turn their own blank into a barrel, just to specific dimensions that you wish?
ER Shaw will build barrels in a wide variety of dimensions and weights.. something they do might be very close to the specs you are looking for... and they are set up to do a wide range of calibers as well... They are pretty backlogged right now with about a 6 month wait... in the past I have had them build barrels for me closer to the 60-90 day time frame..
Shaw Barrels
www.shawcustombarrels.com
For premium barrels, my maker of choice is Shilen.. Ive always gotten outstanding customer service and outrageously accurate barrels from them.. they also make a pretty wide variety of contours and weights, calibers, etc.. you'll pay a bit more for a Shilen.. but if they make what you are looking for.. you wont be displeased.. youre getting the cats pajamas when you get a Shilen..
http://www.shilen.com/contours.html
I'd venture a guess that if you called ER Shaw or Shilen and asked them to build a true custom based on drawings (nothing they currently produce is close enough to what youre looking for) either might be willing to do a one off, custom job for you.. I've spent a good bit of time in the Shilen factory.. they are a big enough shop to have the tools and capability to accommodate about anything.. but small enough that a one off order probably isnt out of the question... But... I'd guess thats going to be a pretty expensive proposition... (but probably at least worth making the call if its something you really want)..
Another option might be to have a barrel maker build you a bull barrel with the chamber and rifling you want.. at that point there are a number of competent gunsmiths that can re-profile that barrel on a lathe to just about anything you want as long as its safe... it would be an expensive solution.. but something I would think would be doable..
For example.. there is a gunsmith I've used over in Plano that would be up to such a task....
he isnt anyone I'd want to actually punch the bore, create rifling, etc.. he just doesnt have the tools or the expertise to do that.. but finishing out a short chamber, cutting a barrel down, adding a crown, or even re-profiling a "heavy" barrel to something lighter that meets my specs/requirements.. he would be more than willing to do that, has the tools to do it, and I think would do a fine job at it.. He'd likely just charge me a small fortune to do it lol...