In a small saucepan, bring 1/2 cup sugar, the orange zest, basil leaves and 1 cup water to a simmer over low, stirring occasionally, until the sugar dissolves. Let cool; strain into a wide, shallow bowl. Transfer 6 tbsp. of the syrup to a medium bowl; add the strawberries and toss to coat. Cover and chill.
In the bowl with the remaining syrup, quickly dip the ladyfingers. Line the edges of a 9-inch springform pan with 3-inch-high sides with the ladyfingers, arranging upright, rounded sides against the pan. (You may need to trim the ends of the cookies so they stand flat in the bottom of the pan.) Use any leftover ladyfingers to line the bottom of the pan, breaking the cookies to fill.
Pour 2 tbsp. cold water in a small microwavable bowl. Sprinkle with the gelatin; let stand until it softens, about 10 minutes.
Meanwhile, in a large bowl using an electric mixer, beat the remaining 1/2 cup plus 2 tbsp. sugar, the butter and salt until light and fluffy, about 5 minutes. Add the cream cheese in four to five additions, beating well between additions and occasionally scraping down the bowl. Beat in the créme fraîche, orange juice and lime juice.
Heat the gelatin in the microwave on high until dissolved, 10 to 15 seconds. Add to the cream cheese mixture; beat just to blend.
In a large bowl using an electric mixer, beat the heavy cream just until soft peaks form. Fold the whipped cream into the cream cheese mixture. Spoon into the ladyfinger-lined pan, gently pressing the cream cheese mixture into the ladyfingers on the bottom and sides of the pan (it will fill about two-thirds of the pan). Smooth the top. Chill until the ladyfingers soften, at least 3 hours.
Spoon the strawberries and juices on top of the filling. Sprinkle with the pistachios. Remove the pan sides. Cut the cake into wedges between the ladyfingers.