The cake: position oven rack in center of oven; preheat oven to 350°; grease the bottom and sides of two 9-inch round cake pans; dust pans with flour and tap out excess.
Sift together the cake flour, baking powder, and salt into a bowl; whisk to combine; set aside.
In the bowl of an electric mixer, using the paddle attachment, beat the butter at medium speed until cream, about 30 seconds.
Gradually add in sugar; beat at high speed until well blended and light, about 3 minutes.
Scrape down sides of bowl as needed; decrease speed to medium and add the eggs one at a time, beating well after each addition and mixing until blended.
Beat in the lemon zest and lemon juice until blended.
Decrease speed to low and add the flour mixture in three additions, alternating it with the milk in two additions, and mixing just until blended.
Scraped batter into prepared pans, dividing it evenly, and smooth the tops with a rubber spatula.
Bake cakes for 22-25 minutes, until they are golden brown around the edges and a pick comes out clean.
Cool cakes in pans on wire racks for 15 minutes; invert cakes onto the wire racks and let cool completely.
The filling: in a bowl of an electric mixer, using the whisk attachment, beat the heavy cream at high speed until firm peaks form; transfer the whipped cream to a smaller bowl, cover, and refrigerate.
In the same mixer bowl (no need to wash it), using the paddle attachment, beat the cream cheese at medium speed until very creamy, about 2 minutes.
On low speed, beat in the vanilla extract.
Add the lemon curd and beat at medium speed until well blended and smooth, about 1 minute.
Remove the bowl from the mixer stand and, using a rubber spatula, gently fold in the whipped cream until almost completely blended.
Cover the bowl and refrigerate while you make the syrup and buttercream.
The syrup: In a small saucepan, combine the water, lemon juice, and sugar; bring to a boil over med-high heat, stirring occasionally to dissolve the sugar; remove the pan from the heat and set aside at room temperature.
The buttercream: in a bowl of an electric mixer, using the paddle attachment, beat the butter at medium speed until cream and light, about 2 minutes.
At medium speed, gradually add the lemon curd, a large spoonful at a time, then add the salt and beat at med-high speed until creamy and light, about 1 ½ minutes; set aside at room temperature while you assemble the cake.
Assemble cake: using a long serrated knife, trim off the domed tops of the cakes so that they are even; cut each cake horizontally into 2 layers; reserve one of the flat bottom layers for the top of the cake.
Place another layer cut side up on a cardboard cake round or flat serving plate; generously brush with some of the lemon syrup.
Spoon a scant cup of the filling onto the cake and, using a small offset spatula, spread it into an even layer, leaving a ½ inch border around the edge of the cake.
Top with another cake layer and brush it with more syrup; top with another scant cup of filling.
Repeat with another layer, more syrup, and remaining filling.
Brush the cut side of the reserved cake layer with remaining syrup; place the layer cut side down on top of the cake.
Frost the sides and top of the cake with the buttercream.
Serve the cake at room temperature; if it has to be refrigerated, let stand at room temperature for about 45 minutes before serving.