HVAC Furnace - Adding C Wire to Older Wiring Style
You already have a C wire, it's just not called by that name

From a close examination of your furnace's wiring diagram, the fat brown wire it supplies in the thermostat wiring compartment is the C-wire feed from the transformer.  The way your system is wired connects the brown wire in the outdoor-unit cable and the brown wire in the thermostat cable to that brown C-wire feed, so the brown wire that connects to the X terminal on your thermostat is your C wire!  As a result of this, just hook that brown wire up to the C terminal on your new thermostat and call it a day.  (If you're wondering what terminal the blue wire goes to on the new thermostat, it went to the B terminal on the old thermostat, so it goes to the O/B terminal on the new thermostat, by the way.)