Reserve out ¼ cup cream and set aside. Using an electric mixer and medium bowl, beat the remaining ¾ cup heavy cream and ⅔ cup sugar on medium-high speed until soft peaks form, about 2 minutes.
In a stand mixer with a paddle attachment or with the electric mixer using a separate medium bowl, beat 1 pound softened cream cheese at medium-high speed until smooth, about 2 minutes. Add the whipped cream/sugar mixture and beat at medium-low until combined, about 1 minute. Scrape bottom and sides of bowl well with rubber spatula.
Add 1 ½ Tablespoons lemon juice, 1 ½ teaspoons vanilla, and pinch salt and continue to beat at medium-low speed for another minute, scraping the sides of bowl again. Increase speed to medium-high and beat until airy, about 1 minute. Add 1 teaspoon lemon zest at this time and beat just to combine.
Pour the reserved ¼ cup of the heavy cream into a microwave-safe measuring cup. Add 1 envelope gelatin and whisk to combine. Let stand until gelatin is softened, about 3 minutes, then microwave on high power for 10 seconds. Let cool about 30 seconds (you don't want it to sit for too long or it will get lumpy). If a film has formed on the gelatin, stir briskly and remove any excess film. Add gelatin mixture to the filling. Beat on medium until smooth and fully incorporated, about 30 seconds.
Pour filling into crust and, using an offset or icing spatula, spread filling out evenly. Refrigerate for 24 hours.