I want to install a Smart Thermostat to control my AC unit. The current "dumb" thermostat is battery powered and does not use a C-wire (see current thermostat wiring pic). When I look at the wiring diagram for the control board it seems relatively clear that a brown wire coming from the receptacle is the one that should be the C-wire (see pics). It appears that it is pigtailed to a white wire, however, and I am not sure if I should just attach the blue wire (which I plan to use as the C-wire to the thermostat) to this wirenut or separately connect wire the brown and the blue wires and separately cap off the white wire that it is connected to.
Just connect the blue wire in with the existing brown/white wires
You should incorporate the blue wire into the existing junction of the brown and white wires. That way, the blue wire will provide C to the thermostat without interfering with the C wire return from the condensing unit's compressor contactor, something that's very important to have otherwise your air conditioner won't work.