Greetings Brew2day - it appears your carbonation issue was the result of not having enough priming sugar. Without all your exact particulars, roughly speaking, a 1 gallon batch would have used .8oz. of priming sugar. One tablespoon is only a fraction of that amount. I’m guessing 2-3 tablespoons of corn sugar would have been closer. The CO2 in the picnic keg would have only provided enough pressure to move the beer out of the keg and into the glass; not enough for a forced carbonation.
Edit: yes, headspace is critical when carbonating. The less headspace available, the more the beer will absorbe the carbon dioxide. Carbon dioxide, being a gas, will fill the void first. Once filled, pressure begins to build and the carbon dioxide gets absorbed by the liquid.