Hawkeye, I too love SxS's. There can be no more effective tool for upland game hunting than a straight gripped, splinter fore end, double triggered, lightweight SxS. However, the very attributes which make such a gun so successful in the uplands work against you on the clays course.
To build a SxS for target use more weight is required and should be biased a little further forward. A pistol grip and semi beavertail fore end will be a distinct advantage for competition, and this combination cries out for a single trigger (there is no advantage to double triggers for any clay target sport anyway). Unless the stock is custom made for you, an adjustable comb will be a definite advantage.
The problem is that no one mass produces a gun like this. CSMC comes pretty close with the RBL, but it is still essentially a hunting gun. I looked at having a Model 21 custom built for the purpose - let's just say it would not fit in your budget.
It would be possible to take An RBL and add an adjustable comb without exceeding your budget by too much. The weight is right and spare parts are available. It can be had with all of your specified requirements. You could probably get one here and add the adjustable comb and be in the 5 to 6 K$ range.
And therein lies the rub - you can go and buy a fully configured U/O competition gun from Beretta or Browning for at least a thousand less. Likely a couple of thousand less. The dedicated SxS guy is going to have to open his wallet wide to indulge his tastes on the clays field. Not many are willing to do that and hence no mass produced SxS's built as dedicated competition guns.
I wish you luck in your quest and hope you find something that satisfies you. Please let us know how things turn out.
EDIT: Just noticed that CSMC does now make an RBL in a sporting configuration. $7K U.S.
Sharptail