Surface mount transformerless 230v to 3.3v on a small board
Consider the somewhat similar requirements the developers of the Belkin WEMO faced- tight quarters, reasonably high current low voltage supply required, mains input, WiFi communication, and an LED for an HMI. 

Here (Edit: Moved to here) is a teardown of that unit. You can see that the designers opted for a small switchmode power supply (and power output stage) on a separate board (probably two layer) from the high-density digital circuitry (probably multilayer). 

That yellow mylar tape-wapped device on the left is a small switchmode power supply (flyback, almost surely) transformer. Aside from being able to deliver the required current, providing safety-agency acceptable galvanic isolation at that point means that they can use inexpensive switches, and don't need to worry about electrocuting someone through a switch operator (or LCD in your case). Judging from the small size, I'd guess they're using triple insulated wire rather than attempting to meet safety requirements by using tape. 

Note also the extra insulation on the wires between the boards, for safety again.