How to reduce the moisture when making tarte tatin
The reason, I think, that most recipes don't leave a thick caramel and it all goes 'runny' is that it's designed to go into a sort of caramel-y sauce when turned out.

Since in most conventional recipes you par-cook the fruit in the caramel then top with the pastry and bake, the fruit releases a lot of moisture into the caramel and making it runnier. Therefore, if you cook the apples separately (boiling them, say) then drain, add to the caramel and cover with pastry and bake, much less juice should leak out of them.

I would keep the caramel recipe the same as in the recipe but just follow the steps above. I don't think the type of sugar will make any difference other then flavor (although soft brown sugars do have more moisture in them).

Also, since everything except the pastry is cooked, try baking the pastry conventionally (in the oven) then grill it for a few minutes to increase the heat on the pastry directly to avoid it becoming soggy.