Make crust: Preheat oven to 400ºF; mist a 9-by-13-inch baking pan with cooking spray. Place crushed pretzels in bowl of a food processor. Add sugar and melted butter and pulse to combine. Press pretzel mixture into baking dish and bake for 8 minutes. Transfer to a wire rack and allow to cool completely, about 20 minutes.
Make filling: Using an electric mixer on medium speed, beat cream cheese with 3/4 cup sugar until well blended, about 2 minutes. Fold in yogurt and vanilla. Spread over cooled pretzel crust, cover with plastic wrap and chill for at least 2 hours.
In a large bowl, dissolve both packages of gelatin in 2 cups boiling water. Let cool until mixture begins to get syrupy, about 15 minutes. Stir in strawberries and blueberries and pour over cream cheese layer. Chill until firm, 2 to 3 hours.
Whip cream until soft peaks form. Add remaining 1 Tbsp. sugar and continue whipping until stiff peaks form. Cut tart into squares and serve with whipped cream.