current draw would be less than 300 mA
Then I don't think it is on. At 300mA for 3 seconds you need 0.18F, and that is the whole charge in the capacitor. You want more than this because that is when the voltage is zero so double it and you get to 0.4F.
I know you can get super caps but I don't think you can get the current density out of them to sustain 300mA.
You would be much better off with a small rechargeable battery that you float charge and then turn off the whole thing in software / hardware after you detect the main power is gone.