Should I play the full chord with the left hand for Travis Picking even when not all strings are plucked with the right hand?
Two reasons for saying yes. Occasionally, you may pluck the wrong string with your right hand. If there's an appropriate note on that wrong string, it'll still sound in tune. Secondly, strings not played, but which could produce a right note, will vibrate sympathetically in tune with those played. Not a great deal, maybe, but this can add to the richness of your sound as harmonics will add to it. So, if you are not using spare fingers to dampen strings, make whole chord shapes.
This is entirely up to you.
It can be simplest to remember to play the whole chord and just pick your way through it, especially if you are holding the chord for a bar or more. In many Travis picked songs this is the way to go.
If, however, you are rapidly moving through chords, it can be much faster to just fret the notes you require and then move your finger to the next note while plucking one of the other fretted notes, so you always have some fingers stationary and some fingers moving.