version 2 maybe?

interface.ui

@@ -97,7 +97,7 @@
       <object class="GtkImage">
         <property name="visible">True</property>
         <property name="icon_name">edit-clear-symbolic</property>
-        <property name="icon_size">80</property>
+        <property name="icon_size">6</property>
       </object>
     </child>
   </object>
@@ -105,13 +105,17 @@
     <property name="visible">True</property>
     <property name="can_focus">True</property>
     <property name="focus-on-click">False</property>
-    <signal name="clicked" handler="backspace"/>
+    <signal name="clicked" handler="de_popover"/>
     <child>
       <object class="GtkImage">
         <property name="visible">True</property>
         <property name="icon_name">applications-system-symbolic</property>
-        <property name="icon_size">80</property>
+        <property name="icon_size">6</property>
       </object>
     </child>
   </object>
+  <object class="GtkPopover" id="de_select">
+    <property name="relative_to">settings_btn</property>
+    <property name="position">GTK_POS_RIGHT</property>
+  </object>
 </interface>

lightdm-mobile-greeter.py

@@ -20,9 +20,28 @@ class Handler:
         self.greeter.connect("show-prompt", self.show_prompt)
         self.pmpt_label = builder.get_object("prompt_label")
         self.msg_label = builder.get_object("message_label")
+        self.de_select = builder.get_object("de_select")
+        vbox = Gtk.Box(orientation=Gtk.Orientation.VERTICAL, visible=True)
+        self.de_select.add(vbox)
         self.greeter.connect_sync()
         self.conf = Config()
         self.greeter.authenticate(self.conf.user)
+        self.session = self.greeter.get_default_session_hint()
+        sessions = iter([(s.get_name(), s.get_key()) for s in LightDM.get_sessions()])
+        name, key = sessions.__next__()
+        first = Gtk.RadioButton.new_with_label(None, name)
+        first.connect("toggled", self.session_change, key)
+        vbox.add(first)
+        for name, key in sessions:
+            rb = Gtk.RadioButton.new_with_label_from_widget(first, name)
+            rb.connect("toggled", self.session_change, key)
+            vbox.add(rb)
+            if self.session == key:
+                rb.set_active(True)
+        vbox.show_all()
+
+    def session_change(self, rb, key):
+        self.session = key
 
     def on_destry(self, window):
         Gtk.main_quit()
@@ -39,7 +58,7 @@ class Handler:
 
     def auth_complete(self, greeter):
         if greeter.get_is_authenticated():
-            greeter.start_session_sync(None)
+            greeter.start_session_sync(self.session)
         else:
             greeter.authenticate(self.conf.user)
 
@@ -49,6 +68,9 @@ class Handler:
     def show_prompt(self, greeter, text, type):
         self.pmpt_label.set_text(text)
 
+    def de_popover(self, settings_btn):
+        self.de_select.popup()
+
 
 Handy.Keypad()
 builder: Gtk.Builder = Gtk.Builder.new_from_file(

style.css

@@ -43,3 +43,11 @@ hdykeypad > button label{
    font-size: 2em;
 }
 
+popover {
+    background-color: black;
+    font-size: 2em;
+}
+
+popover radiobutton {
+    padding: 1em;
+}