A weekend is two days... Fill the dish before you use, and and Boncuk said earlier use a solenoid from a hopper for the second feeding timed... No big woop.
I guess a cat won't die if it's fed with dry food for two days.
Wet food will pretty much mess up any automatic feeder. The question is: Will the wet food really be dispensed?
However here is another mechanical suggestion:
Regulator clocks are normally driven by a pendulum which converts linear movement into rotation using a tooth wheel with the teeth looking like shark fins.
The pendulum drives a pushrod which advances the toothwheel with every stroke. Mounting that tooth wheel to the rotary dispenser you might use as many food "cells" as there are teeth on the wheel.
Here is an image of a regulator clock.
It's a reproduction of the famous "August Tasche" regulator built in Vienna 1850.
Boncuk, the reason for the use of a circular tray with many compartments is so you can fill the tray once a week and it advances once a day to the next try so you can meter the cat food.
The feeder might be used with a big container from which the food is released in portions being controlled by a timer to keep the shutter open for a predetermined time period.
I used that arrangement to feed fish. Since fish live in a wet environment they weren't upset not to get their usual portion of tubifex worms for some time.
I have a battery powered round tray feeder for cats that works on the rotating lid principle. There are 5 chambers in the round tray and the lid only exposes one at a time. The lid is rotary spring loaded and you wind it one turn to start. The built-in battery-powered timer than momentarily operates a cam that allows the spring to rotate the lid to expose the next chamber at the interval selected.