Hi - Any early congrats!
Really briefly....
I think your budget may be a little light if you would like a custom calender compliation....especially a Perpetual. I would maybe get a vintage JLC perpetual movement (or even a semi-current version) and find a watchmaker to use that as the base, and make a customer case and dial for you.
I would contact the US independants representative who used to moderate this forum....you might find his assistance invaluable!
You might also contact Paul Gerber, Roger Smith or, Kari voutilainen - but I hear that Kari is overbooked for the next few years.
I also think that, with your description, a Journe Calendrier would fit the bill nicely....its in your price range, its from a living maker, its 40mm (ok 2mm smaller than you would like, but 40mm is an appropriate size for current styles), it is a truly unique and rare watch. Chances of running into someone with this watch on are slim to none.
Good luck! and keep us posted!