Line the bottom of an 8-inch springform pan with parchment and freeze for 10 minutes. Meanwhile, cut the pistachio ice cream into pieces and beat in a stand mixer with the paddle attachment on low speed until smooth and spreadable but still frozen, about 30 seconds.
Sprinkle 1 cup of crushed pretzels in the bottom of the chilled pan. Spread the softened pistachio ice cream over the pretzels and tap the pan on the counter several times to remove any air pockets. Sprinkle with another 1 cup crushed pretzels and pat evenly. Spread evenly with the marshmallow cream. Freeze the cake until firm, 1 to 2 hours.
Cut the cookie dough ice cream into pieces and beat in the mixer to soften. Spread on top of the marshmallow cream layer, then sprinkle with the remaining 1 cup crushed pretzels. Freeze until firm, at least 2 hours or overnight.
Frost the cake: Rub the side of the springform pan with a warm kitchen towel to loosen. Remove the ring; invert or slide the cake onto a platter and freeze 15 minutes. Beat the heavy cream with the confectioners' sugar and vanilla until stiff peaks form. Spread over the top and side of the cake. Top with mini pretzels. Freeze until firm, at least 3 hours or overnight.